This is a situation that came up recently in a game I play in. Two player characters were riding an elephant in combat. The elephant was acting as a controlled mount under control of one of the riders, who was higher in initiative order than the second. During their turn, the controlling rider had the elephant move part of its speed and take the Dodge action. They then handed the reins to the other rider and dismounted.
What should the second rider be able to do on their turn? In our game, the second rider had the elephant move the remainder of its speed and didn't give it a new action. I feel this is reasonable, but I'm curious whether the rules can be read to give more specific guidance for situations where the controller of a mount changes during combat. In particular, does the initiative order of the first controller, the second controller, and the mount affect what can happen, and how?