During your move, you can pass through the space of an ally, a creature that has the Incapacitated condition (see the rules glossary), a Tiny creature, or a creature that is two sizes larger or smaller than you.
Another creature’s space is Difficult Terrain for you unless that creature is Tiny or your ally.
You can’t willingly end a move in a space occupied by another creature. If you somehow end a turn in a space with another creature, you have the Prone condition (see the rules glossary) unless you are Tiny or are of a larger size than the other creature.
I do agree with @kenclary that using the "carry" option to move an opponent in front of you should be fine.
I also rule that the distance you can drag or carry a grappled creature is commensurate to the distance moved by the grappler.
Move X feet = Drag X feet
So for example every 5 feet space or square a grappler moves, it can drag or carry a grappled creature an equivalent distance of 5 feet space/square, thus preventing disproportionate ping-pong in and out of an area of effect etc...
By your reasoning it is nearly impossible to drag somebody off a cliff or into a fire without going over the cliff yourself.
Whenever a grappler and a grappled creature are adjacent to a cliff or fire and moves, it could drag or carry it into it.
There could be scenario where it gets impossible if for example the grappler is dragging a grappled creature behind it on a 5 feet ledge where the only possible space it can move is forward but these would be edge case, in most situation where there's space to move, it shouldn't be a problem.
Obviously, a DM should find a way to allow a lot of Fun ideas that the players might have in these scenarios -- it might be a bit silly to tell a player that when two creatures are fighting face-to-face on a 5-foot wide "bridge" that it is impossible for either creature to force the other creature off of the side of the bridge.
But when looking at the RAW, as written we really can't get there.
When you look at every place in the game that uses the word "carry", it is implied that the thing being carried is within your own space. You carry your backpack and other gear, you carry your shield, you carry your spellcasting focus, and so on. If you are Medium and your grappled enemy is Tiny, then it would make sense to carry it (within your own space) while you move instead of stooping down to drag such a small (and light -- perhaps 10 pounds or so) creature along the ground. But if your enemy is also Medium, this isn't really a thing. You reach out and grab hold of his shirt collar with one hand while your sword is in your other hand. You really cannot now lift him up within his own square and begin walking around.
Likewise, the word "drag" in this game implies that you are pulling something along the ground behind you -- such as a sled or a corpse or whatever. In this case the thing being dragged remains in the square behind you while you move.
The word "push" in this game implies that you are pushing something in front of you or away from where you are facing. The grappling rules never use this word, but the Shove attack rules DO use this word.
As always, once we understand what the rules actually say we can then choose when to break them. When the situation calls for it, this is one of those rules that should probably be broken in the interest of Fun. As always, remind the players that whenever a house rule benefits a PC against an enemy then that same rule might benefit the enemy against a PC later on.
If you had someone grappled, I would allow the grapple spin but reduce the grappler's movement as if they had moved that distance - unless, it is a monster whose stat block provides for doing so for free.
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Playing D&D since 1982
Have played every version of the game since Basic (original Red Box Set), except that abomination sometimes called 4e.
Interesting. I was under the impression that moving into their space was not a legal move. Cool.
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This is basically what the rule for moving through other creatures say (and it's the same as the SAC answer).
I do agree with @kenclary that using the "carry" option to move an opponent in front of you should be fine.
Whenever a grappler and a grappled creature are adjacent to a cliff or fire and moves, it could drag or carry it into it.
There could be scenario where it gets impossible if for example the grappler is dragging a grappled creature behind it on a 5 feet ledge where the only possible space it can move is forward but these would be edge case, in most situation where there's space to move, it shouldn't be a problem.
And as others have said, such ledge scenario could always instead carry the grappled creature in front of the grappler anyway.
Obviously, a DM should find a way to allow a lot of Fun ideas that the players might have in these scenarios -- it might be a bit silly to tell a player that when two creatures are fighting face-to-face on a 5-foot wide "bridge" that it is impossible for either creature to force the other creature off of the side of the bridge.
But when looking at the RAW, as written we really can't get there.
When you look at every place in the game that uses the word "carry", it is implied that the thing being carried is within your own space. You carry your backpack and other gear, you carry your shield, you carry your spellcasting focus, and so on. If you are Medium and your grappled enemy is Tiny, then it would make sense to carry it (within your own space) while you move instead of stooping down to drag such a small (and light -- perhaps 10 pounds or so) creature along the ground. But if your enemy is also Medium, this isn't really a thing. You reach out and grab hold of his shirt collar with one hand while your sword is in your other hand. You really cannot now lift him up within his own square and begin walking around.
Likewise, the word "drag" in this game implies that you are pulling something along the ground behind you -- such as a sled or a corpse or whatever. In this case the thing being dragged remains in the square behind you while you move.
The word "push" in this game implies that you are pushing something in front of you or away from where you are facing. The grappling rules never use this word, but the Shove attack rules DO use this word.
As always, once we understand what the rules actually say we can then choose when to break them. When the situation calls for it, this is one of those rules that should probably be broken in the interest of Fun. As always, remind the players that whenever a house rule benefits a PC against an enemy then that same rule might benefit the enemy against a PC later on.
If you had someone grappled, I would allow the grapple spin but reduce the grappler's movement as if they had moved that distance - unless, it is a monster whose stat block provides for doing so for free.
Playing D&D since 1982
Have played every version of the game since Basic (original Red Box Set), except that abomination sometimes called 4e.
The Grappling rules don't say the grappler has to be in the same space as the grappled creature to carry it but a DM can always rule this way.