You shoot those targets about 35 yards ±15 away from where you are standing. You'll hear an alert (or call) for a target which will appear somewhere out in front of you about 3 seconds later. They may be traveling quickly or slowly, near the ground or high above you, in the same or opposite directions. You'll have about 5 seconds to hit two targets with two shells from a "double barrelled" shotgun.
I've little doubt that with an afternoon of practice I could reliably hit clays moving 30% faster; not as accurately as in an ordinary game, but certainly well over 50%.
If the targets were larger and need to be "clipped" rather than fully destroyed then the task becomes easier. Coupled with a shotgun that can hold more than 2 shells, modern sights, and semi-automatic action: I think reliably shooting targets like that is possible.
If drones become considerably more durable then you'd need a tighter pattern of heavier shot to delivery sufficient energy to disable or destroy it. Tighter patterns would make hitting the target more difficult as you need to be more accurate when aiming. As the size and power behind your shot increases it also becomes more difficult to make follow-up shots if you miss because the recoil pulls you off target.
The biggest hinderance wouldn't be speed but maneurverability. In order to hit a moving target you need to be able to predict where it will be, if a drone moves erratically then it would be very hard to predict and so very hard to shoot.
Drones on the other hand, once they become more intelligent, should be able to fly in a random pattern, to avoid incoming projectiles fired against it.
A rifle round like a Nosler 26 will travel a quarter mile with no drop. Fired out of a long barrel, with a good rifling twist, accuracy is ridiculously good. There are plenty of high powered rounds that don't have to be fired with nearly as much prediction necessary in skeet -- because theyll be traveling at 4k feet per second. If the drone is within a mile..its a piece of cake.
If I remember the details correctly, the Metal Storm[1] weapon system should be able to do just that: Fire explosive rounds set to detonate at a specific point in their flight. Fire n^3 rounds to cover enough space to account for speed, direction and altitude changes and you should be set.
