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.
While I agree with the answer of MrNattious, consider that the OP asked for what happens when "multiple PCs grappling the same target NPC?" Often when this happens it is because the PC's are attempting to help one another move together, so I think it may be valuable to point out how the PC's can act so as not to "get left behind", if that is their goal.
It is likely that only one PC can move the grappled creature on their turn and that the other PC's, absent some other action, will then get left behind - if the movement removes the grappled creature from their reach, the grapple will be automatically broken. In order to prevent this, they will need to move with the grappled creature when it is not their turn.
Acting when it is not one's turn is done through readied actions. It is important to understand that a readied action can be used to take an action, or move, but not both. If the goal of the PC's is to move the grappled creature with all of the PC's maintaining their grapple, some or most of them should ready actions to move with the grappled creature when any one of them moves the creature.
Consider four PCs (1-4) and a creature (C). Suppose the turn begins with no readied actions, with all four of the PC's grappling the creature, and with the initiative order 1, 2, C, 3, 4.
In a simple case, PC's 1, 2, and 3 could, on their turns, declare that they were readying an action to move with the grappled creature when it moved or was moved. PC 4, as the last in the initiative order, could then on their turn move the grappled creature, triggering the readied actions of the other three and allowing them to come along without breaking their grapples even if the distance moved was greater than their reach.
In a more complicated case, the PC's could realize that on the grappled creature's turn, it could make an attempt to escape the grapples1, and if it was successful, their readied actions would then have been 'wasted' as they could move with it but would have lost their grapples. Thus strategic players might decide that the PC's going before the creature should ready actions to protect the grapple, while PC's going after the creature should ready actions to move with it. Assuming the same initial conditions and turn order as before, one possible iteration of this might be:
PC1: Readies an action to Help a PC with their contested roll, should the creature attempt to escape a grapple.
PC2: Readies an action to re-establish a grapple, should the creature succeed in breaking their grapple.
PC3: Readies an action to move with the creature, when it moves or is moved.
PC4: Uses its action to move, and moves the still-grappled creature along with them.
1It is an open question of whether one action can be used to break multiple grapples, but that is my opinion and there is some unofficial support for that.
Strictly RAW, It goes pretty much like Kirt says above.
Some ready their move, one does the moving, at half rate dragging the target, the others move along, maintaining their grapples. The target on his turn may try to break the grapples, and that would work like most such group activities -- the best grappler rolls with advantage since the others are helping maintain the grapple.
There is no provision in the rules for everyone helping drag the target, as there is no way to "help" with movement, RAW. So group speed is limited to 1/2 the speed of the active last grappler. If the grapplers have different speeds, then some juggling of readied actions would be needed to have the fastest one's speed count for dragging the target, but it could be done.
In my world, they could all help move the guy faster. I would run it like this:
The normal dragging means that the grappler is spending half his movement on his own move, and also spending half his movement on moving the target. If some or all of the grapplers all readied their moves to help, I would let them split the amount spent to move the target between them.
So, for example, suppose everyone has 30 movement and there are 4 grapplers, 3 of whom have readied their actions to move -- splitting 4 movements (120' of movement points) among 5 people -- they all can move and drag the guy 24 feet (120' move / 5). Playing on a grid, I'd round that to five 5' squares.
(I also in my world modified it a bit so that dragging something larger than you takes double, and something smaller takes half. So one medium creature dragging a medium creature can go 15' [15' grappler and 15' spent to move target]; medium dragging a large can get 10' [10' grappler and 20' spent to move target 10'], and a medium dragging a small can get 20' [20' grappler and 10' spent to move target 20'].
In your multiple-grappler scenario, I'd adjust the amount it costs to move the target accordingly, and split that among the grapplers.
Most DMs probably won't want to do that much math on the fly, and would run it differently, but it works for me.