N Medium humanoids have gathered on an arbitrarily long field, standing in a line with each creature 120 feet away from each of its neighbors. The creature at one end of the line is mounted on a horse with a speed of 60. The creatures are arranged in initiative order, such that all N creatures will act in the order of their position in a single melee round. The mounted creature acts first.

Each creature other than the mounted creature is holding a pike and has readied an action to Shove the next mounted creature that moves adjacent to it.

On its turn, the mounted creature commands its mount to move 60 feet towards the next creature, then drops prone (which is not stated as a condition for dismounting in the Mounted Combat section of the 5e PHB) and orders the horse to Dash and move 60 feet more, closing the distance. At this time, the next creature's readied Shove triggers, and it makes an attack on the rider. Assuming the attack hits (which is likely, given that the rider is prone, thus granting advantage, and for the sake of the example has no armor or dexterity bonus to AC) the rider will be pushed off of his mount and the mount leaves his control. At this time he has used his actions and the second creature's turn begins.

Creature 2 starts by mounting the now-adjacent horse, whose initiative changes to match his. *However*, the horse's turn ended when its first rider's turn did, and thus it is taking a new turn when its second rider mounts it. The second rider orders it to move 60 feet towards the third creature, drops prone, and orders it to Dash another 60 feet, triggering the third creature's readied Shove. This procedure continues until one of the Shove attacks fails, allowing the horse to move a theoretically infinite distance in a single 6-second round.

I am absolutely certain there is a flaw in this logic, but I can't find it – what error breaks this strategy?