This question is an example of how a DM should deal the concept of Rules Lawyer vs Playability in the game.
Given the way the rules are written, there's a loophole that can be exploited. Yes, it's possible to string those actions together.
However, logically, take a minute to think about what you ask. In a round (which is 6 seconds), can you: move 120', change mounts, then move another 120', after which the rider then makes some attack?
The fastest horse in the world can do 2 furlongs (1320 ft) in 20.6 seconds, which translates into 384' movement rate. That's at full gallop, not from a standing start. Even presuming the character is going to leap between saddles of moving horses as the dismount/mount action, adding an attack at the end of the round stretches believability beyond the breaking point.
Good DM-ing in D&D is not just about adhering solely to the letter of the law; this will absolutely fail, because there's no possible way to write rules that cover all possible situations, and questions like yours frequently pop up when defined actions result in unintended outcomes. Logic and consistency is the path to take, and being free to ignore the strict letter of the rules is a key part of being a good DM. Not capriciously, but when the rules clearly aren't rational. This is such a case.
TL;DR: Yes, it's possible, according to the letter of the law. NO, a DM should not allow this kind of rules exploit. Under no circumstance should additional movement be allowed after the character takes their full movement themselves, unless they decided to forgo any other actions in the round.