“Asylum” is basically a magic word used in an attempt to gain access to the US. These folks didn’t go to a port of entry and ask for asylum because they are being threatened by the government, etc… They wandered across the border and then, when caught, cry asylum. These aren’t, for example, Jews fleeing Hitler’s regime or Stalin’s pogroms. They aren’t former residents of Iran from before the Islamic theocracy, members of the “White Russians” during the revolution, or similar.
They are people who have wandered in for economic reasons, just like the folks who came to the US from Europe. The difference is those folks wound up going through ports of entry and went through the process to become citizens. They didn’t just wander in.
With regard to ensuring that the children are actually theirs, would almost have to DNA test the whole lot of folks.