SpringingIntoAction "British living in many EU countries already have to apply for residence after set periods of time."
Again, this is incorrect. As a national of the EU, we currently have a right to reside in any EU country provided that we can satisfy the conditions of being a worker, self employed, student or self-sufficient. Once you have been resident in an EU country for 5 years whilst holding status in one or more of those categories you acquire a permanent right of residence. However, there is no obligation to make any sort of application for recognition of that status. So there are a significant number of people who hold a right of permanent residence but have no stamp in their passport to prove it.
A pragmatic view will be to transfer everyone with a permanent right of residence onto some other category of indefinite leave to remain, but this is not a given in any country. It would be perfectly open to any country to require people prove, for example, that they are still economically active above a particular threshold.
It is a particular rose-tinted view to think that there will not be thousands of people in both directions who lose the right to reside in their host country of choice, and to think that people will not suffer real infringement to their private and family lives as a result.