"Learning" when test and train distributions don't match
In general, the results are pretty strongly negative --- fairly strong assumptions are needed for something like this to work. As an extreme case, suppose that training and testing distributions have disjoint supports, so that the training sample can be nearly uninformative regarding the test performance.

That having been said, there is a body of research on learning (with) changing target concepts/distributions. Here are four fairly representative samples; once you do a follow-up search of who cites them and who they cite, you'll pretty much span the space:

https://link.springer.com/chapter/10.1007/978-3-642-34106-9_13

https://link.springer.com/article/10.1023/A:1007604202679

https://dl.acm.org/citation.cfm?id=269812

https://link.springer.com/article/10.1023/A:1007666507971

Also potentially relevant: Transfer learning
https://en.wikipedia.org/wiki/Transfer_learning