I never thought about it, but that might work. If we just sent them their welfare checks to their Mexican address, they might decide to stay in their own country. After all, they would live better there. The American dollar is still worth something in Mexico.
"Would anyone be surprised if we were that stupid?"
At first look, it seems stupid, but lets give it some thought.
I'm sure some figured out some way to keep receiving them. An aunt of a woman I know just was arrested for receiving welfare benefits in 4 different states by using different combinations of several of her actual names and false documents but she's out of jail already. She was legalized in the 1980's mass amnesty program, she's worried she might get kicked out of the US but that probably won't happen.