Yes. Their circumstances shouldn’t matter. They have the right to ask, nicely, to enter (which implies WE HAVE a right to say no) but they have no right to just impose themselves.
The only process they should be due is having to pay the bill for their detention and quick return.
No need for detention for those who come through an airport, bridge, or road. Just immediately have them turn around. What we need for most of the rest of the border are a double electric fence on our border and a small open area between the fences for cbp K-9 units. For those caught in the interior then yes it’s more complicated. In all cases any country of direct origin that doesn’t take back people and/or cargo without legal entry permission should immediately accept them back. If they refuse then we close down all direct access. For example, if Canada doesn’t take them back then people traveling to the USA from Canada will need to go through third party countries such as the Bahamas, Mexico, or uk until border normalization can be restored.
Let them ask at the nearest US Embassy/Consulate in their ancestral homes. They’ve been taught that all they have to do is get feet dry, north of the Rio Grande and they’re good.
Toss in a kid and it’s like winning a scratch off ticket.
Cross the river, right before their water breaks, pop out a citizen and they’ve hit the lottery.