@JeremyECrawford while atop a controlled mount, can you grapple a creature and drag it around with the mount's movement? Or would the mount's movement end the grapple?
— [OO]DataBlue (@DataBluePS2) October 3, 2018
The rules detail how to move a creature you're grappling by using your own movement. If you try to move the creature some other way, you need to use the rules on lifting and carrying.
— Jeremy Crawford (@JeremyECrawford) October 3, 2018