Requires the library mod Collective.
This mod is part of The Vanilla Experience.
Move Minecarts is a drag-and-drop mod which allows players to easily pick up and move minecarts without breaking them by holding right click.
You may also be interested in Move Boats.
mustSneakToPickUp (default = true): When enabled, will only pick up the minecart when the player is crouching and holding right-click.
To pick up: Hold right-click and crouch while looking at the minecart. You can release the mouse button while moving the minecart.
To release: Right-click the minecart again.
Here you can see a minecart being picked up; be careful with a tnt cart!