I personally support much more open immigration, as do many other libertarians. When I get push-back from my libertarian friends, it generally is on two fronts:
- You can't combine open immigration with a welfare state -- this leads to financial implosion
- Open immigration allows illiberal, anti-democratic people to take power through the democratic process (a phrase I stole from here, though it is not actually about immigration). In the name of liberty, we let people come in and vote for authoritarian illiberal measures.**
I agree that both of these are real problems. The key for me is to disassociate legal presence in this country from citizenship. It should certainly be possible to have multiple flavors of legal presence in this country. At level 1, anyone can legally be present, seek employment, buy property, and have access to certain services (e.g. emergency services). At level 2, history of working and paying payroll and income taxes gets more access to welfare-state sort of programs. Over time, this may or may not lead to full citizenship and voting rights, but there is no reason we can't still be careful with handing out full citizenship while being relatively free with allowing legal work and habitation.
** I have observed a US internal version of this. People run away from California to places like Arizona and Texas to escape California's dis-functionality But as soon as they arrive in their new home, they start voting for the exact same crap that sank California.