DM:
Upon further review, I've found clarification from the powers that be at WoTC that "monsters" don't follow the same grappling rules in the PHB, so I'll adjust my post to only be a grapple vs. Morg. Good catch!
My understanding is that when you've grappled something, when you move you can drag or carry the
grappled creature with you, but your
speed is halved unless the creature
is two or more sizes smaller than
you. It's not a shove, just moving while grappling something else. Since these are monks and not special monsters, they should probably use the same rules as PCs.