@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.
As DM, I'd allow your mount to move at half speed so that you could maintain a grapple. #DnD https://t.co/ndbZ8WHi4U
— Jeremy Crawford (@JeremyECrawford) October 3, 2018