This won't happen, at least not the way you expect.
When you're at maximum capacity, any new recruit will land in the Waiting Room. Mother Base will of course try to optimize staff assignment by replacing old staff with new one. Those who are least suited for a team will be reassigned to one where they can replace someone even less suited to it, or land in the Waiting Room if there aren't any.
However, for some of your staff, Mother Base won't ever reassign them. Those are staff members with unique skills. If you have a low-rank Russian interpreter, he will remain in your Support team unless you specifically assign him somewhere else. If you happen to have 2 Russian interpreters, the lower-ranked one will be reassigned, while the other will remain in your Support team. This isn't limited to interpreter skills and also extends to R&D-relevant ones, like Bionics Specialist and more. These people won't Auto Assign, even if you ask for it. You'll have to manually assign them to a specific team.
People who just left the Brig will be assigned to the teams best suited to them, but they won't replace anyone, even if they'd be better suited to it. Instead they'll land in the Waiting Room until you choose to assign or Auto Assign them.1
Finally, people who you have manually assigned to a team will never be auto assigned by Mother Base, unless you manually assign or Auto Assign. Staff that are under a Direct Contract can only be assigned manually, and won't ever automatically change their team, even if you ask for an Auto Assign.
The Waiting Room is quite spacious and offers a lot of space for staff. Making use of that room is useful for when you plan on expanding Mother Base and do not wish to part with your A-ranked staff, just because they happen to be lowest ranked.
However, the Waiting Room does not offer an unlimited amount of space. I am currently unsure how many people fit in it and if the size of Mother Base affects that limit. When I hit it, there were 175 people waiting inside. I am also unsure if that limit can be exceeded by assigning staff to it manually.
What happens when the Waiting Room is full, along with the rest of Mother Base? Some of the people waiting inside will automatically be dismissed. Who exactly is chosen, I don't know. It could be the oldest recruit, or it could be the lowest ranked one, or it could also be chosen at random.
1: There are time when people who leave the Brig will replace less suited staff in full teams, and times when that doesn't happen until you specifically request an Auto Assign. I haven't yet been able to pin down when it happens and when it doesn't, so I've decided to strike this point out for the time being.