It's correct that people moved around since forever. But I think applying the the concept of illegal immigration to colonization of Americas doesn't make sense.

The indigenous Americans didn't organize their societies as nation states.

So, I doubt that they had "borders" in the sense used here. Supporting this is the fact that their cartography wasn't very advanced. Here's how their maps looked like: https://www.archaeology.org/issues/338-features/maps/7549-ma...