One reason migrants anywhere get paid less is not economics but exploitation: They have limited power, including legal rights and limited access to the legal system to enforce those rights. Unfortunately, others exploit them.

> If you raise the minimum wage high enough so that it provides for a liveable wage in a major city and are able to enforce it, then the amount of people willing to pay migrants for work will collapse and the migrants will stop coming to the city.

While I understand the theory, the reality in places that raise minimum wage is much more complex.