U.S. Geological Survey map (fair use policy)
"Mature" rivers like those of the American midwest do a lot of meandering. The natural tendency of a river flowing across flat country is to to grow more and more curvy, forming meanders or loops as it erodes the outer side of a bend and deposits sediment on the inner side. It's clear that several of Red River's loops have changed this way since the border was fixed in 1841.
This portion of the Geologic Map of Arkansas shows what happens when a permanent boundary is based on a temporary thing. In 1841 the border between the Republic of Texas and the United States was surveyed along Red River, as shown by the heavy dashed line. Specifically, the boundary was set on the southern bank of the riverin the words of a Supreme Court decision, "the water-washed and relatively permanent elevation or acclivity at the outer line of the river bed which separates the bed from the adjacent upland."
The survey was very carefully done, and the results of the survey, as recorded in the engineers' report and monuments placed along the border, were accepted by both governments as the true and legal boundary. But the river paid no mind, and in the years since 1841 the Red River has left the border high and dry. It's usually more trouble than it's worth for two governments to move a boundary. And so it stays.
Often the river breaks through the base of a loop (the process is called avulsion), cutting off the meander, and leaves the curve behind as an oxbow lake. The small lakes on the map above are all oxbow lakes, sitting where the river once flowed. The yellow lobes south of the river are more recent examples of meander cutoffs. The map to the right shows two clearer examples of avulsion from Arkansas's east border at the Mississippi River.
The wide zone of yellow color shows where the river has laid down sediments this way during Holocene time. It's clear that riverbeds can move around quite a lot over geologic time.