ride (someone or an animal) down
to chase down someone or an animal while riding on horseback. The mounted policeman rode the mugger down and captured him. The rider rode down the thief. We had to ride down the runaway horse.
ride something down
to ride on something that is going down, such as an elevator. You take the stairs, and I will ride the elevator down. I don't want to ride the cable car down. I will walk.