I have a player who loves to just grab other PCs or NPCs to get them out of danger in the middle of combat. He's got the highest DEX mod in the party so he's usually at the top of the initiative ahead of everyone else.
For example, there is a damsel in distress who acts later in the initiative and the party is being attacked by thugs. They're on the second floor, so he ties off a rope in the first round, then stands between her and the thugs to prevent them from attacking her (it's a small room), then on the next turn, he grabs her around the waist and jumps out the window with rope in hand.
I'm fine with the idea, but every time it happens a part of me cringes because it feels like it's circumventing the initiative order. In the scenario above, if the damsel is the target of the thugs, and the thugs would have acted before her, it removes the opportunity for further conflict to occur because his first instinct is to grab her and run.
To be clear, the above is simply one example. I'm not concerned with that scenario specifically, so a good answer should address the following concern:
How can a PC move a willing ally without violating movement rules (by granting another PC or NPC "extra" movement) or allowing out-of-initiative movement?
How can I handle this in a way that doesn't affect his agency as a player but also doesn't "cheat" the system by essentially allowing free out-of-turn movement? Am I worrying about nothing? I want to encourage creative actions such as this as it's far more interesting than a slugfest to protect the girl, but I also want to adjudicate the rules and not give an unfair advantage to the players.
Three steps to deal with this, although step zero is probably “yes, don't worry about it so much.”
Make sure the action is actually possible first. A character only gets one free simple interaction with an object, such as closing or opening a door, or drawing or sheathing a weapon, or jumping up on a table. Pulling out a rope (interaction #1), tying it around a post (#2), grabbing someone (action), and jumping out the window (interaction #3 maybe? + movement) is more than can be accomplished in one turn. And carrying a fellow adult — even a cooperative one — is non-trivial, so make sure that's not happening for free either: pull in the encumbrance rules and penalties on movement speed to see if they can actually jump out that window this turn, or if they maybe end their turn perched precariously on the windowsill, just in time for the thugs to get their turn.
Also note that if the hero is physically blocking the thugs over two turns in order to fit all the interactions in, the thugs will naturally be adjacent to him and will get opportunity attacks on the hero (but not the damsel) when he moves out of their reach by jumping out the window. He also sometimes won't effectively block them if he's busy mucking about with a rope unless it's a very narrow space, because you need two hands for tying rope; thugs can just move past an unarmed hero without provoking opportunity attacks.
If he grabs her arm and “leads her away”, then that's not going to work at all — she can't follow his lead without using her own movement, so all he's doing is Readying an action to walk with her when it's finally her turn. (He would have to grab her arm and drag her away, not using her own movement, to make it happen on his turn. And I'm sure the damsel will have words with him after — or during — such a maneuver…)
But however it works out, let it work out how it should by judicious leverage and application of the rules rather than trying to decide that it shouldn't happen at all.
Notice that forced movement is a thing in the game and doesn't break any rules. Initiative is just who gets to act first; it doesn't prevent others later in the initiative count from benefiting from those actions.
Is this an interesting way to make one character end up farther away than they could on their own? Yes, but it's at the expense of another character spending their turn in a rather sub-optimal way. (Although, sometimes the real goal is better served by doing something “sub-optimal”, so the definition of that is arguable.) Pushing and pulling others around is something you can do with your action, so there's no reason to be alarmed about it being used to improve an ally's position.
This doesn't end the conflict. Just because he's moved an ally out of imminent danger doesn't mean the danger packs up and goes home. He jumps out the window with her? Well… quite apart from the risk of dropping someone because you're jumping out a window and trying to hold up the weight of two people with bare hands and no climbing harness (which you might want to talk about, and maybe call for checks regarding), you still have a conflict — just not a combat.
There's nothing stopping the thugs from peering out the window, shouting “Follow them!” and clambering down the rope too; or dropping rocks, chairs, cannonballs, and priceless bric-a-brac out the window on our intrepid heroes' heads; or hustling down to ground level to kick off a breathless chase conflict.
So yes, don't worry about this. Roll with it, take any necessary few moments to consider how to apply the rules to adjudicate the attempt, find out if it works, and then make the enemy react realistically and interestingly to the unexpected tactic.