I recently noticed a question posted to MO that was migrated to MSE. Furthermore the comment was also migrated along with the question. The content of the comment was "This is not the right site for your question", which is correct on MO but not on MSE. I am sure that the poster of the comment did not realize this and therefore cannot be faulted. That said, I think this is something to be aware of when considering migrations. The question mentioned is https://mathoverflow.net/questions/135827/calculate-the-derivative-using-fundamental-theorem-of-calculus?noredirect=1 .
The question has "migrated from ..." banner on the target site, so it is usually pretty obvious that such comments were written on the source site. Users can just flag them and they'll be cleaned up.
Most questions need some cleanup anyway after migration, the tags usually don't fit exactly and there are some obsolete comments most of the time.
Perhaps the etiquette should now be to write comments with the expectation that such questions will be migrated. For example, a comment could be: "This question is better suited for math.stackexchange, rather than mathoverflow. I recommend it be migrated to MSE."