I don’t see migration increasing without limit, or even in the short term as e.g.
May has suggested:
we may very well see in the run-up to that, people wanting to come here to the UK before that exit happens
That strikes me as nonsense. People aren’t going to suddenly rush here on the off chance they will be unable to do so after an exit. They are going to wait and see, or delay making plans, so if anything migration will reduce.
Migration from the EU can’t keep growing forever, and is likely to fall sharply, even go into reverse, at some point. Growth in the main countries people are coming from + a recovery in the rest of Europe will reduce the incentive to leave and give them many more places to go. New countries joining could make a difference, except that is not likely any time soon.
Migration from outside the EU is a problem, but not for us with our largely effective border measures, and as we are outside the Schengen area. It might yet cause greater rifts within the EU, though they seem to have found a solution for now with the deal with Turkey.