From what I've heard the reason for the hatians flooding springfield is some factory wanting them for cheap labor, if we knew who the factory was maybe an email campaign or boycott could be useful.
Maybe a don't work with scummy factory campaign as opposed to a don't work with scummy magazines campaign
No, they wouldn't. Because most people aren't willing to move thousands of miles to a foreign country with a different language to get shit pay.
Why wouldn't you voluntarily work at a McDonald's in Dead Horse, Alaska? You've got no barriers. No one is stopping you. You could buy a plane ticket right now. There's free movement within the US. So why don't you?
Because the cost of that kind of a move is so fucking crazy both economic, culturally, and personally, that you literally wouldn't ever consider it. Neither do most other people.
A free market would mean that the government is NOT subsidizing this shit.
Your decision might change if the government promised you free food, free shelter, free healthcare, free travel, and a bonus. Most people still wouldn't take it because most people are invested in their local environment by one mechanism or another (career, family, culture, familiarity, etc). The only people who would take that are rootless people, mercenaries who will keep no investment, or criminals trying to get away.
That is why it works the way it does.
The answer to your hypothetical is if Dead Horse, Alaska can't draw anyone to work at McDonald's, it doesn't have a McDonald's and that is an absolutely acceptable outcome. We don't need to import an army of third-worlders to go staff a fast food restaurant somewhere that clearly isn't established enough to justify one's presence. If someone living in Dead Horse really wants to live near a McDonald's, they are the ones who need to exercise free movement.
If no American wants to go there to work at a McDonald's there and lack of garbage food isn't enough to make the current residents leave, then you have an equilibrium. The market is working just fine. Maybe some people are sad they can't have a Big Mac, but that is fine.
Hence my point. Even a completely lax and open border in a purely free market doesn't equate to hordes of people flooding in. You kill the subsidy system and the whole migration racket goes with it.