I did not think I was going to actually find a professional research paper on this, but I will happily take it. If you look at a map, a lot of the political borders you see- state lines, national borders- are human-made. The straight lines, the zig-zagging from land grabs and ethnic separations. But other borders are purely geographical: the ridge of a mountain range, following a river, or the coastline (or middle) of a particular body of water.
This generally works pretty well, because nobody's expecting a mountain to move itself anytime soon. But rivers do move. Have you ever sent a little stream of water from a faucet down a flat surface, maybe a pan? You ever see streams of rain dance around on your windshield? Rivers do that, just a lot more slowly because they have all that dirt to dig through. They'll make little burrows into a patch of ground somewhere and go thataway instead of (or in addition to) the thisaway they'd been going beforehand. Older rivers will show this because they'll have a ton of little islands showing the places where the river changed direction over the years.
This happens on a timescale short enough to where, if you live on a border designated by a river, it is entirely possible that one day you will wake up one day to find yourself on the other side of the border. Because the border follows the river. As NPR illustrates, one place that has seen a particularly large amount of grief over this is the Rio Grande, separating El Paso, Texas and Ciudad Juarez, Mexico. The river's changed course multiple times since the Civil War, and at one point in the 1960's caused a forced relocation of the residents of a region called the Chamizal, which was occupied by El Paso residents but which the river said was now part of Ciudad Juarez.
The river has since been encased in concrete. Not that everybody thinks that'll be a permanent solution.
Internally within the United States, this comes up once in a while, but the stakes aren't quite that high. The borders between Texas and Oklahoma, and between Georgia and Tennessee, have had this come up in recent years, but when it's between states, it can often be more about the pride of having the land than any kind of thing they plan to do with it.
And as John W. Donaldson of Durham University in the UK notes (there's that paper I mentioned!), there are rivers in other countries too, and their moving is often going to make for a national border dispute. Some settle the dispute by arranging to lock in one particular line as the boundary and no longer caring what the river thinks of it. But as this might reduce one nation's access to water, that's not always the solution sought. It depends on what's more valued, the land or the water. It can end in some agreement or other, it can end in arbitration by the International Court of Justice, or it can just simply not end at all and spur bouts of occasional violence between the locals.
Not that the river cares.