It's not super easy to emigrate and the systems countries have set up won't be capable of handling x times what they currently process. So it's fair to say many more people could be trying to leave but people leaving doesn't actually rise much.

A lot of the Europeans left although that's a bit of a different thing really - them going home rather than Brits leaving. I know a few Brits who have left although it's not especially down to Brexit, more it just being a bit dreary here these days. Though Brexit has contributed to the dreariness.