It kicks off at Dover all the time. In the past it has mostly been about industrial action by the French workers. And being smart strategists, they always find the worst time to do it.
Yes I'm sure Brexit adds some new problems/delays in terms of passport checking, but they aren't huge. Great get out to blame though by various other responsible parties. After all the ferry companies aren't exactly going to pass up the chance of a perfect faceless scapegoat.
Passport checks, I remember once at Dover being given a good going over long before Brexit. They went through every page in my passport, stopped on a few that had weird visas and had a conversation with me about each one. Then said to me that I must get a lot of hassle, to which I replied not normally I have two passports and I usually travel on the other one and they just waved me through. I've also been though Schipol in the last few days and went straight through without them so much as glancing at the photo ID, so not sure that they check everything ultra thoroughly, although airports obviously are a bit more controlled with respect to who is in them and where they are from in the various zones.
They may be a bit more thorough with kids because of events that have happened in the past, after all I'm pretty sure no parent on here would want the border authorities waving through kids at random without proper checks, and those checks take time.
I wouldn't go within 50 miles of Dover near a bank/start of the school holidays - just seems like a recipe for disaster to me.