I'm sure Schengen or an equivalent will continue to exist. All they will do it put border controls in place - right now you can drive from France, through Belgium and into The Netherlands and zoom up to Germany at 100mph. No controls whatsoever. (I used to live in Amsterdam and once managed to cross Belgium by car in 40 minutes.) In future due to this immigration/refugee issue they will add checkpoints. If you have a valid Schengen visa (which is like a valid Schengen passport) then on you go.

Today even the borders with Switzerland are loosely policed. Rose and I drove down there last week and I only stopped to get the vignette (motorway tax paid sticker). Nobody checked our passports. Same happened back in 2009 when I was driving my 'anti-social' TVR with my Dad in the car.