From the story here quoting Matthew Albence of ICE:
"I will say that for the vast majority of the individuals [who leave their children]… it is because they completed the smuggling act. Their goal when they paid their smuggler, the criminal organizations, these cartels, $5,000 or $6,000 to smuggle themselves into the country, their goal was to get their child here. They’ve accomplished that goal. So if they return on their own [to their home country], they are willing to do so and leave the child here because that was the intended goal of their illegal entry in the first place." ...
DHS officials also cannot force parents who are released from detention in the United States to pick up their claimed children from the shelters operated by the Department of Health and Human Services, said Albence. “If individuals choose not to do that, there is nothing we can do to force them to claim their child,” he said during a telephone interview arranged by DHS.
Foreign children and youths who are left in the United States by their parents are allowed to file for green cards on the grounds that their parents abandoned them. The migrant children are also eligible for various welfare and aid programs, and are allowed to attend taxpayer-funded schools and universities. Once adopted by the United States, then legalized and naturalized, the new Americans will be allowed to bring in their foreign parents and siblings via unreformed chain-migration laws.