They can figure out a way to get people to Venezuela faster than due process, they can figure out a way to get a British nanny back to London without detaining her indefinitely.

They could of had her home within 48 hours.

how about denying them entry, and turning them back to the country they were entring from (Mexico in your example), so that they can fly back home from there?

That sounds reasonable, though I’ve read online that Mexico generally refuses to take them back.