Watching the film, it struck me that these groups of random strangers who had paid to travel from Missouri to Oregon would be together, in relatively close quarters, with no interruption, in a high-stress situation, with low comfort, for many months. Surely there must have been many disputes, akin to disputes neighbors get into today. How were such disputes resolved? Were temporary property rights enforced? If so, how and by whom? Was there a kind of "wagon train arbitration system" or an HOA of sorts?
Eugene Ware goes into this in a little detail in his book The Indian War of 1864 from when he was a lieutenant in the Nebraska Territory.
He says that the wagon trains had a boss who was often the most dangerous man in the group and frequently carried a pistol. His primary job was to guard against theft in the train. Some of the convoys could reach 900 wagons, with thousands of people and animals, so keeping an eye on the merchandise would be quite a task. Punishments for infractions could include banishment from the train.
In the case of unscrupulous bosses or disputes between different wagon trains, the U.S. Army would act as an arbiter when the trains were in the proximity of a fort. They could enforce payments and incarcerate troublemakers and those who couldn't pay debts. Sentences occasionally included locking up a particularly troublesome person until the train was several days underway, so that the culprit would be unable to rejoin the procession.
The military command at that time in that region included a lot of men who were lawyers in their civilian careers. These felt that the role of the military should be to maintain the rule of law in the lawless plains. As such, they would dispatch soldiers to retrieve reported criminals in the trains.
Once trains got out of range of a fort it was up to the wagon-master to maintain order.