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.