I think the issue is sending the people to Mexico when they aren't from Mexico.
If someone refuses to say their country of origin, how do you know where to send them?
And that's where Mexico needs to step in and do something about this border bullshit on their end to keep it from happening. Immigration has become a joke lately anyway. The way things were under Bush, if we arrested somebody for something as simple as publix intoxication, and he didn't have his papers, his ass was gone as long as we were willing to do a little paperwork. ICE would come interview them, determine if they were legal or illegal, and take them with them if they were illegal. They would then be held until they had a deportation hearing, which usually happened within a few weeks. Within 4 weeks of their arrest, Juan Doe would be back in Mexico.
The way it works now, if they are arrested, ICE determines their immigration status, they go to Mobile where they are given a court date for 8-12 months later, released, and never show up for court. Do you think they are going out and hunting them down when they fail to appear? I doubt it. Doing it this way just creates a whole new problem. We used to have a shit-ton of people deported. We've called ICE once since everything changed. It's just not worth it to do all the paperwork and make the effort when they aren't actually doing something about it.