The Rulers of these dying States don't want any Libertopians moving in and giving their new neighbors dangerous ideas. Thus the immigration "controls". I doubt it's necessary- who would move to a State? But it was a good justification for putting the infrastructure in place to control the "borders".
Because, even more, they want to keep the residents (the States would call them "citizens") they have from escaping- it's hard to tax people who leave if they go somewhere without a government to help you track and control them. It is also hard to rule when you are alone.
Most of the residents of these States were in favor of the immigration controls. At least at first. What the residents of these States failed to realize was that any policy or agency that can keep "them" out, can be turned around and be used to keep "us" in. Cages and "borders" work both ways.
Libertopians living near these unilaterally-declared "borders" are usually more than willing to help people escape an oppressive State (Yeah, I'm repeating myself there). The more people who are free, the better it is for everyone. It is the new "Underground Railroad", and it is killing the last few States.
It is funny, though. No one in Libertopia worries about immigrants moving in and using up all the welfare. There is no welfare. No one worries about immigrants taking all the jobs- they are just as likely to create jobs. No one worries about what language anyone else speaks. There can be no "official language"- you communicate however you can. No one worries about their culture being destroyed by exposure to another culture. That would be an admission that your culture isn't worth saving. Yet all these excuses, as unbelievable as they sound to me, are actually used in those dying States!
Without a government to co-opt or take over- either by voting or by having more offspring than someone else- people don't seem to worry too much about who is moving where. At least as long as the new neighbors don't initiate force or steal. And if they do, where they came from is the least important piece of information you could possibly worry about.