I'm not sure that immigration policy is relevant to libertarianism because a nation in some sense is like private property. So one could argue that the people of a country have a collective right to restrict who enters their borders. Transversing a nation's airspace would be a different story. I think if a nation blocked other nations from using its airspace for transversal it would be a violation of other people's rights, by abusing, essentially, private property exclusivity.

That's just a way to rationalize policies that are obviously anti-liberty. Is the Texas/Mexico border my property? Really? All of it? And isn't abuse of private property anti-liberty anyway? You literally just said it is.

Countries do prevent other countries from using their airspace, by the way.

Preventing people from encroaching on your nation is fundamentally no different than preventing people from trespassing on your property. It's not anti-liberty, since we don't have an inherent right to any land on earth. That right to occupy a piece of land, to the extent that it exists, emerges through homesteading and the principle of First Possession.

As for U.S. territory, yes, you can make a case that it's the collective property of American citizens who then decide how the property will be governed through their elected representatives. How is that anti-liberty?

Through what right do countries exist and have rights and those rights transfer to their citizens? What causes me to have a right to control the Texas/Mexico border?

A political community can make a claim over a territory because it builds institutions that build and defend it. It's not identical to homesteading, but it's similar. Think of private property and national sovereignty as different layers of ownership that emerge through similar principles.