Can one PC grab another PC and drag them?

While DMing my session today, I encountered a situation I wasn’t sure how to answer. The PCs were attacked by a werewolf and decided to flee onto their nearby horses. Two of them were already mounted and two of them were on foot. One player on the ground (a wizard) attempted to cast a spell on the werewolf and failed. He then moved closer to the mounts but couldn’t make it with his speed. After a few turns it was the barbarian’s turn. The barbarian wanted to drag the wizard with him and plop the wizard onto a horse and then mount with his remaining movement speed.

I wasn’t sure quite how to handle it, so I made a compromise. I allowed the barbarian to drag the wizard. However, because the rules specifically state that it takes half of a character’s movement to mount, I wouldn’t allow the barbarian to force the wizard to mount. However, if the barbarian wanted to, he could mount the horse himself as he had enough remaining movement.

The reason I allowed the barbarian to drag the wizard is because of the lifting, dragging, and pushing rules in chapter 7 of the Player’s Handbook. The barbarian had more than enough strength to push our tiny, lightweight wizard around.

I’m not really sure if I handled it right at all. I’m not entirely sure that those rules apply to living beings, especially living beings that can move. I would really appreciate some advice on whether or not players can drag other players after their (the player who is being dragged) turn is over or if players can drag other players at all. If I recall correctly, I believe the wizard still had about 5′ of movement left, if it makes a difference.