A lot of these children were separated from their parents while their parents were seeking asylum. So no, it's not a forgone conclusion that the child will be cared for in the other country.
Hell, in some cases they're not even sure which country the child is from.
If they cannot determine where a child is from, I am sure that the US government will take care of the child.
If they can, and the asylum request is approved, then the child stays in the US.
If they can, but the request is denied, then their home country is not an unsafe place, and surely the home country will want their citizen back.
I feel like talking about asylum is an edge case in its self. Surely not the majority of immigrants are not legitimate asylm seekers. For most, they may not like their home countries (perhaps due to the low wages, or opportunity) but home is not a place of abuse where we must morally rescue them from.
Hell, in some cases they're not even sure which country the child is from.