According to the news channels here, a lot of people can easily cross the border once we leave the eu because we don't have many treaties. I recently went to do a Google search as I thought it was odd since the number of wars we had in the 11th century and our alliance in wars also and found this list
https://en.m.wikipedia.org/wiki/Entente_CordialeIn which at least 8 treaties are related to the uk, more than the news articles suggested.
I don't think it's ever expressly stated that they won't start randomly trying to send people over who aren't residents (unless it's the second to last treaty - the eu one which we've pulled out of almost). Generally it's not included in treaties as anyone who does it is a victim of having another country overtake them. If 2 million migrants from the eu gained citizenship since the last election, we could technically be forced to stay in the eu if another referendum were called.
Ultimately, leaving the eu should make the uk more secure from migrants, I and a lot of other people have managed to get across the border without a passport check over a decade ago - we were with a British company but there isn't any security when getting on the vehicle. (I had a passport on me and wasn't asked for it).
In other news, the British government has given 13+ million to a company to improve a port without any trading history - posted on Christmas Eve so they could hide it. The plans for it to fail are going well.
I also read two major collapsing German banks are being joined by the government (or planned to be anyway) so it seems we might be leaving the sinking ship at the right time.