When a Battle Master uses their Maneuvering Strike maneuver, does the creature they're affecting have to take the reaction immediately after the attack?
(PHB pg 74)
Maneuvering Attack. When you hit a creature with a weapon attack, you can expend one superiority die to maneuver one of your comrades into a more advantageous position. You add the superiority die to the attack’s damage roll, and you choose a friendly creature who can see or hear you. That creature can use its reaction to move up to half its speed without provoking opportunity attacks from the target of your attack.
Here's an example:
You have a Battle Master fighter named Dave and he is riding a warhorse and wielding a lance.
Dave holds his action to attack with his lance as soon as there is an enemy within ten feet.
On the horse's turn, it runs toward an enemy, and Dave hits the enemy with an attack and makes it a maneuvering attack, which he gives to his horse.
Does the horse then have to use its reaction to gain the benefits of the maneuvering attack?
Instead, can it wait until it has gotten into melee with this creature, take its attack, and now use its reaction to get out without provoking an attack of opportunity? (I'm aware Dave will still provoke AoO).