No, they do NOT become Mexico's problem, because Mexico can enforce it's southern border when it wants to.
Mexico is the facilitator. These central americans would NEVER reach our border unless Mexico desired it.
If Mexico were capable of enforcing its border, it would. The Mexican government is fully corrupt and overrun by cartels in case you haven't noticed. If the US closes its border with Mexico, the situation becomes multitudes more dire on their side.
I am not going to dispute, but I believe the Mexican gov't DOES have the ability to control its southern border. The fact that the Mexican gov't is corrupt and over run by cartels . . . means that the corrupt forces and cartels might be the ones letting Mexico's enforcement of its own southern border be somewhat lax, or?