For shooting Trap or Skeet, most shooter have a " range " box, like to two men in the background loading at the table in the picture above. The shot and powder are kept in the box, along with the wads, and lube. Often the box is set up specifically to carry a large chamber that holds the shot. A dipper is used to move the required shot from the box to the barrel. Powder is usually measured with an adjustable measure at the bench. I have seen as many Trap shooters loading from a can, with a spout on the cap, as load from powder horns. Caps are kept in a capping tool, and extra caps are in the box. I have seen skeet shooters use boxes where the percussion caps are carried open in a small box in their range box, and they take the needed caps in their hands as they go to each station to shoot. It all depends if they are using a DB shotgun, or a single barrel gun. You see more shooters with single barrel guns carrying that cap in their hand, than those who shoot double guns, who use a capper, instead. Obviously, all the cleaning patches and solvents are in the box, too.
I see a lot of styrofoam wads being used as cushion wads to solve the problem of weighty cushion wads following the shot, at these events. They get a wipe of lube around their outside edges before they are pushed down the barrel. Because the Styrofoam will NOT absorb lube, they remain pretty light, and drop to the ground about 10 yards in front of the gun. As long as they are lubed, before being pushed down the barrel, I am told they do not leave plastic residue in the barrel.
As for cleaning, I often see Trap shooters with two range rods at their loading stations- one has a bore brush on it, and the other a normal cleaning jag. The brush is used after each shot to break loose the lead deposits, and the cleaning jag is used in cleaning and loading the smoothbore. I don't see the brushes present with Skeet shooters as much, as they shoot at much targets that are much closer to the muzzle.