They were all part of the RA, as Kevin said. And the RHA were formed from the RA exactly in order to increase the speed of deployment of the guns.
You might find some interesting footage of King`s Troop of the RHA deploying their guns in Hyde Park to fire salutes on the Queens Birthday.
Both foot and horse artillery used horse teams to move their guns and limbers. In India it was common to use teams of bullocks to move guns, sometimes even elephants.