diff --git a/src/DataStructures/CMakeLists.txt b/src/DataStructures/CMakeLists.txt
index 1396ffa167a7..4cd42f9e3af9 100644
--- a/src/DataStructures/CMakeLists.txt
+++ b/src/DataStructures/CMakeLists.txt
@@ -41,6 +41,7 @@ spectre_target_headers(
   DynamicBuffer.hpp
   DynamicMatrix.hpp
   DynamicVector.hpp
+  ExtractPoint.hpp
   FixedHashMap.hpp
   FloatingPointType.hpp
   IdPair.hpp
diff --git a/src/DataStructures/ExtractPoint.hpp b/src/DataStructures/ExtractPoint.hpp
new file mode 100644
index 000000000000..1f004c5273d3
--- /dev/null
+++ b/src/DataStructures/ExtractPoint.hpp
@@ -0,0 +1,112 @@
+// Distributed under the MIT License.
+// See LICENSE.txt for details.
+
+#pragma once
+
+#include <cstddef>
+
+#include "DataStructures/DataVector.hpp"
+#include "DataStructures/Tensor/Tensor.hpp"
+#include "DataStructures/Variables.hpp"
+#include "Utilities/ErrorHandling/Assert.hpp"
+#include "Utilities/Gsl.hpp"
+#include "Utilities/TMPL.hpp"
+
+/// \ingroup DataStructuresGroup
+/// Copy a given index of each component of a `Tensor<DataVector>` or
+/// `Variables<DataVector>` into a `Tensor<double>`, single point
+/// `Tensor<DataVector>`, or single-point `Variables<DataVector>`.
+///
+/// \note There is no by-value overload extracting to a
+/// `Tensor<DataVector>`.  This is both for the practical reason that
+/// it would be ambiguous with the `Tensor<double>` overload and
+/// because allocating multiple `DataVector`s for the return type
+/// would usually be very inefficient.
+///
+/// \see overwrite_point
+/// @{
+template <typename... Structure>
+void extract_point(
+    const gsl::not_null<Tensor<double, Structure...>*> destination,
+    const Tensor<DataVector, Structure...>& source, const size_t index) {
+  for (size_t i = 0; i < destination->size(); ++i) {
+    (*destination)[i] = source[i][index];
+  }
+}
+
+template <typename... Structure>
+Tensor<double, Structure...> extract_point(
+    const Tensor<DataVector, Structure...>& tensor, const size_t index) {
+  Tensor<double, Structure...> result;
+  extract_point(make_not_null(&result), tensor, index);
+  return result;
+}
+
+template <typename... Structure>
+void extract_point(
+    const gsl::not_null<Tensor<DataVector, Structure...>*> destination,
+    const Tensor<DataVector, Structure...>& source, const size_t index) {
+  ASSERT(destination->begin()->size() == 1,
+         "Output tensor components have wrong size: "
+         << destination->begin()->size());
+  for (size_t i = 0; i < destination->size(); ++i) {
+    (*destination)[i][0] = source[i][index];
+  }
+}
+
+template <typename... Tags>
+void extract_point(
+    const gsl::not_null<Variables<tmpl::list<Tags...>>*> result,
+    const Variables<tmpl::list<Tags...>>& variables, const size_t index) {
+  result->initialize(1);
+  expand_pack((extract_point(
+      make_not_null(&get<Tags>(*result)), get<Tags>(variables), index), 0)...);
+}
+
+template <typename... Tags>
+Variables<tmpl::list<Tags...>> extract_point(
+    const Variables<tmpl::list<Tags...>>& variables, const size_t index) {
+  Variables<tmpl::list<Tags...>> result(1);
+  extract_point(make_not_null(&result), variables, index);
+  return result;
+}
+/// @}
+
+/// \ingroup DataStructuresGroup
+/// Copy a `Tensor<double>`, single point `Tensor<DataVector>`, or
+/// single-point `Variables<DataVector>` into the given index of each
+/// component of a `Tensor<DataVector>` or `Variables<DataVector>`.
+///
+/// \see extract_point
+/// @{
+template <typename... Structure>
+void overwrite_point(
+    const gsl::not_null<Tensor<DataVector, Structure...>*> destination,
+    const Tensor<double, Structure...>& source, const size_t index) {
+  for (size_t i = 0; i < destination->size(); ++i) {
+    (*destination)[i][index] = source[i];
+  }
+}
+
+template <typename... Structure>
+void overwrite_point(
+    const gsl::not_null<Tensor<DataVector, Structure...>*> destination,
+    const Tensor<DataVector, Structure...>& source, const size_t index) {
+  ASSERT(source.begin()->size() == 1,
+         "Cannot overwrite with " << source.begin()->size() << " points.");
+  for (size_t i = 0; i < destination->size(); ++i) {
+    (*destination)[i][index] = source[i][0];
+  }
+}
+
+template <typename... Tags>
+void overwrite_point(
+    const gsl::not_null<Variables<tmpl::list<Tags...>>*> destination,
+    const Variables<tmpl::list<Tags...>>& source, const size_t index) {
+  ASSERT(source.number_of_grid_points() == 1,
+         "Must overwrite with a single point.");
+  expand_pack((overwrite_point(make_not_null(&get<Tags>(*destination)),
+                               extract_point(get<Tags>(source), 0), index),
+               0)...);
+}
+/// @}
diff --git a/src/DataStructures/Tensor/CMakeLists.txt b/src/DataStructures/Tensor/CMakeLists.txt
index a549370ebb54..a80b062d22cc 100644
--- a/src/DataStructures/Tensor/CMakeLists.txt
+++ b/src/DataStructures/Tensor/CMakeLists.txt
@@ -5,7 +5,6 @@ spectre_target_headers(
   ${LIBRARY}
   INCLUDE_DIRECTORY ${CMAKE_SOURCE_DIR}/src
   HEADERS
-  ExtractPoint.hpp
   Identity.hpp
   IndexType.hpp
   Metafunctions.hpp
diff --git a/src/DataStructures/Tensor/ExtractPoint.hpp b/src/DataStructures/Tensor/ExtractPoint.hpp
deleted file mode 100644
index 470786c1e8b2..000000000000
--- a/src/DataStructures/Tensor/ExtractPoint.hpp
+++ /dev/null
@@ -1,22 +0,0 @@
-// Distributed under the MIT License.
-// See LICENSE.txt for details.
-
-#pragma once
-
-#include <cstddef>
-
-#include "DataStructures/DataVector.hpp"
-#include "DataStructures/Tensor/Tensor.hpp"
-
-/// \ingroup DataStructuresGroup
-/// Copy a given index of each component of a `Tensor<DataVector>`
-/// into a `Tensor<double>`.
-template <typename... Structure>
-Tensor<double, Structure...> extract_point(
-    const Tensor<DataVector, Structure...>& tensor, const size_t index) {
-  Tensor<double, Structure...> result;
-  for (size_t i = 0; i < result.size(); ++i) {
-    result[i] = tensor[i][index];
-  }
-  return result;
-}
diff --git a/src/Evolution/DgSubcell/SetInterpolators.hpp b/src/Evolution/DgSubcell/SetInterpolators.hpp
index 6125c6947010..fd514325633d 100644
--- a/src/Evolution/DgSubcell/SetInterpolators.hpp
+++ b/src/Evolution/DgSubcell/SetInterpolators.hpp
@@ -8,8 +8,8 @@
 #include <optional>
 #include <utility>
 
+#include "DataStructures/ExtractPoint.hpp"
 #include "DataStructures/FixedHashMap.hpp"
-#include "DataStructures/Tensor/ExtractPoint.hpp"
 #include "DataStructures/Tensor/Tensor.hpp"
 #include "Domain/Creators/Tags/Domain.hpp"
 #include "Domain/Domain.hpp"
diff --git a/src/Evolution/Systems/GrMhd/ValenciaDivClean/FixConservatives.cpp b/src/Evolution/Systems/GrMhd/ValenciaDivClean/FixConservatives.cpp
index 1d23f71360ce..d59364e9e3b7 100644
--- a/src/Evolution/Systems/GrMhd/ValenciaDivClean/FixConservatives.cpp
+++ b/src/Evolution/Systems/GrMhd/ValenciaDivClean/FixConservatives.cpp
@@ -10,9 +10,9 @@
 #include <pup.h>  // IWYU pragma: keep
 
 #include "DataStructures/DataVector.hpp"
+#include "DataStructures/ExtractPoint.hpp"
 #include "DataStructures/Tags/TempTensor.hpp"
 #include "DataStructures/Tensor/EagerMath/DotProduct.hpp"
-#include "DataStructures/Tensor/ExtractPoint.hpp"
 #include "DataStructures/Tensor/Tensor.hpp"
 #include "DataStructures/Variables.hpp"
 #include "NumericalAlgorithms/RootFinding/TOMS748.hpp"
diff --git a/tests/Unit/DataStructures/CMakeLists.txt b/tests/Unit/DataStructures/CMakeLists.txt
index 695767b6e2dd..5d387be44b8e 100644
--- a/tests/Unit/DataStructures/CMakeLists.txt
+++ b/tests/Unit/DataStructures/CMakeLists.txt
@@ -16,6 +16,7 @@ set(LIBRARY_SOURCES
   Test_DataVectorBinaryOperations.cpp
   Test_DiagonalModalOperator.cpp
   Test_DynamicBuffer.cpp
+  Test_ExtractPoint.cpp
   Test_FixedHashMap.cpp
   Test_FloatingPointType.cpp
   Test_IdPair.cpp
diff --git a/tests/Unit/DataStructures/Tensor/CMakeLists.txt b/tests/Unit/DataStructures/Tensor/CMakeLists.txt
index 4fc0e087a5ab..6e152a01037e 100644
--- a/tests/Unit/DataStructures/Tensor/CMakeLists.txt
+++ b/tests/Unit/DataStructures/Tensor/CMakeLists.txt
@@ -4,7 +4,6 @@
 set(LIBRARY "Test_Tensor")
 
 set(LIBRARY_SOURCES
-  Test_ExtractPoint.cpp
   Test_Identity.cpp
   Test_Metafunctions.cpp
   Test_Slice.cpp
diff --git a/tests/Unit/DataStructures/Tensor/Test_ExtractPoint.cpp b/tests/Unit/DataStructures/Tensor/Test_ExtractPoint.cpp
deleted file mode 100644
index daaf189bf37d..000000000000
--- a/tests/Unit/DataStructures/Tensor/Test_ExtractPoint.cpp
+++ /dev/null
@@ -1,35 +0,0 @@
-// Distributed under the MIT License.
-// See LICENSE.txt for details.
-
-#include "Framework/TestingFramework.hpp"
-
-#include "DataStructures/DataVector.hpp"
-#include "DataStructures/Tensor/ExtractPoint.hpp"
-#include "DataStructures/Tensor/Tensor.hpp"
-
-SPECTRE_TEST_CASE("Unit.DataStructures.Tensor.ExtaractPoint",
-                  "[DataStructures][Unit]") {
-  const Scalar<DataVector> scalar{DataVector{3.0, 4.0, 5.0}};
-  CHECK(extract_point(scalar, 0) == Scalar<double>{3.0});
-  CHECK(extract_point(scalar, 1) == Scalar<double>{4.0});
-  CHECK(extract_point(scalar, 2) == Scalar<double>{5.0});
-
-  tnsr::ii<DataVector, 2> tensor;
-  get<0, 0>(tensor) = DataVector{1.0, 2.0};
-  get<0, 1>(tensor) = DataVector{3.0, 4.0};
-  get<1, 1>(tensor) = DataVector{5.0, 6.0};
-  {
-    tnsr::ii<double, 2> expected;
-    get<0, 0>(expected) = 1.0;
-    get<0, 1>(expected) = 3.0;
-    get<1, 1>(expected) = 5.0;
-    CHECK(extract_point(tensor, 0) == expected);
-  }
-  {
-    tnsr::ii<double, 2> expected;
-    get<0, 0>(expected) = 2.0;
-    get<0, 1>(expected) = 4.0;
-    get<1, 1>(expected) = 6.0;
-    CHECK(extract_point(tensor, 1) == expected);
-  }
-}
diff --git a/tests/Unit/DataStructures/Test_ExtractPoint.cpp b/tests/Unit/DataStructures/Test_ExtractPoint.cpp
new file mode 100644
index 000000000000..c2f6b9545cd1
--- /dev/null
+++ b/tests/Unit/DataStructures/Test_ExtractPoint.cpp
@@ -0,0 +1,118 @@
+// Distributed under the MIT License.
+// See LICENSE.txt for details.
+
+#include "Framework/TestingFramework.hpp"
+
+#include <array>
+
+#include "DataStructures/DataBox/Tag.hpp"
+#include "DataStructures/DataVector.hpp"
+#include "DataStructures/ExtractPoint.hpp"
+#include "DataStructures/Tensor/Tensor.hpp"
+#include "Utilities/Literals.hpp"
+#include "Utilities/TMPL.hpp"
+
+namespace {
+struct ScalarTag : db::SimpleTag {
+  using type = Scalar<DataVector>;
+};
+
+struct TensorTag : db::SimpleTag {
+  using type = tnsr::ii<DataVector, 2>;
+};
+
+template <typename... Structure>
+Tensor<DataVector, Structure...> promote_to_data_vectors(
+    const Tensor<double, Structure...>& tensor) {
+  Tensor<DataVector, Structure...> result(1_st);
+  for (size_t i = 0; i < result.size(); ++i) {
+    result[i] = tensor[i];
+  }
+  return result;
+}
+}  // namespace
+
+SPECTRE_TEST_CASE("Unit.DataStructures.ExtractPoint",
+                  "[DataStructures][Unit]") {
+  // Test with owning Tensors as well as Variables to make sure there
+  // are no assumptions about memory layout in the tensor versions.
+  const Scalar<DataVector> scalar{DataVector{3.0, 4.0}};
+  tnsr::ii<DataVector, 2> tensor;
+  get<0, 0>(tensor) = DataVector{1.0, 2.0};
+  get<0, 1>(tensor) = DataVector{3.0, 4.0};
+  get<1, 1>(tensor) = DataVector{5.0, 6.0};
+  Variables<tmpl::list<ScalarTag, TensorTag>> variables(2);
+  get<ScalarTag>(variables) = scalar;
+  get<TensorTag>(variables) = tensor;
+
+  Scalar<DataVector> reconstructed_scalar{DataVector(2)};
+  tnsr::ii<DataVector, 2> reconstructed_tensor(DataVector(2));
+  Scalar<DataVector> reconstructed_scalar_from_dv{DataVector(2)};
+  tnsr::ii<DataVector, 2> reconstructed_tensor_from_dv(DataVector(2));
+  Variables<tmpl::list<ScalarTag, TensorTag>> reconstructed_variables(2);
+
+  const auto check_point = [&](const size_t index,
+                               const double scalar_component,
+                               const std::array<double, 3>& tensor_components) {
+    const Scalar<double> expected_scalar{scalar_component};
+    tnsr::ii<double, 2> expected_tensor;
+    get<0, 0>(expected_tensor) = tensor_components[0];
+    get<0, 1>(expected_tensor) = tensor_components[1];
+    get<1, 1>(expected_tensor) = tensor_components[2];
+    const Scalar<DataVector> expected_scalar_dv =
+        promote_to_data_vectors(expected_scalar);
+    const tnsr::ii<DataVector, 2> expected_tensor_dv =
+        promote_to_data_vectors(expected_tensor);
+    Variables<tmpl::list<ScalarTag, TensorTag>> expected_variables(1);
+    get<ScalarTag>(expected_variables) = expected_scalar_dv;
+    get<TensorTag>(expected_variables) = expected_tensor_dv;
+
+    {
+      Scalar<double> result{};
+      extract_point(make_not_null(&result), scalar, index);
+      CHECK(result == expected_scalar);
+    }
+    {
+      tnsr::ii<double, 2> result{};
+      extract_point(make_not_null(&result), tensor, index);
+      CHECK(result == expected_tensor);
+    }
+    CHECK(extract_point(scalar, index) == expected_scalar);
+    CHECK(extract_point(tensor, index) == expected_tensor);
+    {
+      Scalar<DataVector> result(1_st);
+      extract_point(make_not_null(&result), scalar, index);
+      CHECK(result == expected_scalar_dv);
+    }
+    {
+      tnsr::ii<DataVector, 2> result(1_st);
+      extract_point(make_not_null(&result), tensor, index);
+      CHECK(result == expected_tensor_dv);
+    }
+    {
+      Variables<tmpl::list<ScalarTag, TensorTag>> result(1);
+      extract_point(make_not_null(&result), variables, index);
+      CHECK(result == expected_variables);
+    }
+    CHECK(extract_point(variables, index) == expected_variables);
+
+    overwrite_point(make_not_null(&reconstructed_scalar), expected_scalar,
+                    index);
+    overwrite_point(make_not_null(&reconstructed_tensor), expected_tensor,
+                    index);
+    overwrite_point(make_not_null(&reconstructed_scalar_from_dv),
+                    expected_scalar_dv, index);
+    overwrite_point(make_not_null(&reconstructed_tensor_from_dv),
+                    expected_tensor_dv, index);
+    overwrite_point(make_not_null(&reconstructed_variables), expected_variables,
+                    index);
+  };
+  check_point(0, 3.0, {{1.0, 3.0, 5.0}});
+  check_point(1, 4.0, {{2.0, 4.0, 6.0}});
+
+  CHECK(reconstructed_scalar == scalar);
+  CHECK(reconstructed_tensor == tensor);
+  CHECK(reconstructed_scalar_from_dv == scalar);
+  CHECK(reconstructed_tensor_from_dv == tensor);
+  CHECK(reconstructed_variables == variables);
+}