The border is international territory; therefore it is under the domain of the Feds.
I'm not sure that's true.There's a town in Northern Vermont that I pass through fairly regularly (Derby Line) that has an interesting history.It's right on the border.I've stepped right up to the little concrete "monument" indicating the border.No sign of my being in "international territory".I was standing in US territory...if I had taken one step forward I would have been in Canadian territory.
There are parts were private land is right up against the border and those land owners are protected by the 5th Amendment.
The Texas/Mexico border runs right down the middle of the Rio Grande. There are times and places where the forces of nature render it under the domain of no one lesser than the Almighty.
The land on this side of the border is not; therefore it belongs to individuals, Indian tribes, and states for the most part.
A goodly portion of that border is underwater, namely the Rio Grande.