Monitoring machine learning models in production
The changes in distribution with respect to training time are sometimes referred to as concept drift.
It seems to me that the amount of information available online about concept drift is not very large. You may start with its wikipedia page or some blog posts, like this and this.
In terms of research, you may want to take a look at the scientific production of João Gama, or at chapter 3 of his book.
Regarding software packages, a quick search reveals a couple of python libraries on github, like tornado and concept-drift.
Update: recently I came across ML-drifter, a python library that seems to match the requirements of the question for scikit-learn models.