So, lets imagine that the problem is being caused by other states sending their homeless over. Well, in the US you can't stop that, at least by direct migration, you might be able to stop particular organizations like city governments from doing so. But in a general sense there is freedom of movement.

The problem comes with that, the better you're at solving homelessness the more people will get sent there, and even more homeless will want to come there even if they are not sent. Suddenly Texas, for example would see California spending billions on the problem... Texas would have no motivation of spending billions to solve the problem themselves, in fact in terms of getting rid of the appearance of a homeless problem it would be more beneficial for them to be draconian on homelessness. This would push individuals to migrate to the promise land of California.

California would see is homeless costs increase dramatically and would be incentivized to stop (increasing?) funding on the problem themselves. Moloch wins. More people suffer. Yay humanity.