How to update a labeling layer when datasource has changed?

Well, at the moment, the easiest way is to make a join from label layer to source layer based on a common primary key (hope you have one), and just use field calculator to update the fields you need.

If there are too much changes, it can be easier to create a new labeling layer, and then join the new labeling layer with the old labeling layer to get back the old labeling parameters you need (XY in general).

If this is too heavy, why not thinking of an embedded function linking source layer and labeling layer and check for updates. Please ask if the need occurs...