Doing things to encourage a self-deport would help.
I like to look at the problem from unusual angles too.
Why in the world is Mexico (say) WANTING to export people? (and yes, they do... they openly print literature encouraging it). That doesn’t say a lot for Mexico. Could Trump sit down with those guys and say that as long as you keep doing this, it is the worst of both worlds for both of us? You’re killing your home morale, believe it or not, by treating your people that cheap. And you’re saddling us with an impossible welfare burden.
Essentially, Mexico as it is presently run cannot support its population. There is some evidence that some of the exportees are undesirables. (Criminals, the difficult to employ or keep employed, etc.) The majority however are decent enough people. Many are fleeing poverty, the drug wars, etc. They include many hard workers who send home lots of money. "Win-win" for Mexico.