When a PC has someone grappled they are forced to move where the PC moves, I don't have the exact rules wording handy, I apologize. How does this interact with multiple PCs grappling the same target NPC? When one of the PCs move do all 3 participants move, does the other PC get left behind? Ideally I'd like the RAW answer, but I'm open to an interpretation as well.
As far as I can tell, the RAW about this would be: if one of the PCs that is grappling the NPC decided to move the NPC, they can do so without restriction but if they move the NPC out of the other PCs reach, the Grappled condition imposed by those PCs ends, since as the rules for the Grappled condition (PHB p.290) state:
The condition ... ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when the creature is hurled away by the thunderwave spell.
Now, granted, dragging someone away is a little different from being hurled away by a blast of concussive force, but the condition places no restrictions on any kind of forced movement. So thus, dragging someone away from other grapplers would break their grapple (if you dragged them out of their reach).
Finally, an interpretation at my table: if the PC were to do it without their party's consent, I'd allow for a contested strength check between all PCs involved. Failed check would mean no dragging away.