This is Texas.. The White part of the map is privately owned property. The Light Green bits belong to the Forrest Dept, The Dark Green bit belongs to the Fed Gov.
Just looking at the original land grants ~ none of them are INTACT. They've been subdivided and sold off to others.
But, never mind that the property lines of any piece of land in America are subject to existing law. I didn't see that any of the Mexican landgrants extended beyond the Rio however, so nobody had a right in Mexico.
Just doing some genealogy this weekend and found that an ancestor with just about half the gold in Europe decided he wasn't happy with his property lines between his little place in Brittany and adjacent French territory.
He obtained a privilege from the King of France to buy some of that land and did so giving him an estate (with thousands of acres of prime artichoke land) that crossed the border.
He was then able to buy materiel in France cheaper than in Brittany and not pay taxes. He could also deliver his vegetable crops to France without customs as well.
Looked like a great deal then they had a war.
That's when he hightailed it to Sweden to try his luck.
What you imagine goes on at national borders simply isn't the same as what happens at the state line. The United States has authority along the border that is not subject to Texas law!