The problem with sending "judges" is that the "courts" they hold, will have to held upon U.S. soil.
The goal of border reinforcement is to keep the invaders FROM TOUCHING U.S. soil, because once they do so, they're here. And from that point, it becomes very difficult to "get them OUT OF here".
Don't permit them to come across the border in the first place.
If this means that normal crossing points must be closed, then close them.
Be restrained with the use of force, at least initially.
But don't be afraid to use increasing levels of force, if that's required.
Do we want them to reach here, or not?