dnd 5e – What happens when you have a creature grappled and use the Bait and Switch Maneuver to move 5 feet away from the creature?

The Bait and Switch Maneuver (Battlemaster Fighter feature):

Bait and Switch. When you’re within 5 feet of a creature on your turn, you can expend one superiority die and switch places with that creature, provided you spend at least 5 feet of movement and the creature is willing and isn’t incapacitated. This movement doesn’t provoke opportunity attacks.

Roll the superiority die. Until the start of your next turn, you or the other creature (your choice) gains a bonus to AC equal to the number rolled.

Does it just break your grapple since the creature is outside your reach? Does this count as “Moving a Grappled Target” and the creature you’re grappling moves 5 feet with you?

Or does this count as the latter, but since your speed is halved, and the maneuver only allows you to move 5 feet, your speed is 2.5 feet, and if playing on a grid, actually cannot move?