I personally have not run into this - my upgrade from pre-2014 to post-2014 went 100% smooth for the ones that did migrate and the ones that we haven't migrated yet, we need to do more testing before we pull that trigger.
But I think that your friend here is going to be query store. I believe the recommended thing to do is upgrade to 2014+, turn on query store, capture enough data that most (if not all) stored procedures and ad-hoc queries have been run, then switch compatabiltiy mode and if things slow down, have a peek in query store to see which execution plans changed for the worse and address those queries.
Another thing you can do is run the data migration assistant prior to upgrade which will tell you which things are going to be breaking changes and behavior changes.
If you have any database monitoring tools, those can help too. Several companies make good ones; it just depends on your needs and budget.