It depends about what type of data are we talking: tabular, image, text...
This is part of my PhD, so I am completely biased, I will suggest Explanation Shift. (I would love some feedback). It works well on tabular data.
In the related work section one can find other approaches.
The main idea under "Explanation Shift" is seeing how does distribution shift impact the model behaviour. By this we compare how the explanations (Shapley values) look on test set and on the supposed Out-Of-Distribution data.
The issue is that in the absence of the label of OOD data, (y_ood) one can not estimate the performance of the model. There is the need to either provide some samples of y_ood, or to characterize the type of shift. Since you can't calculate peformance metrics the second best is to understand how the model has changed.
There is a well known library Alibi https://github.com/SeldonIO/alibi-detect
That has other methods :)