If you really train with a pump you can keep up with an auto. I've shot pumps for everything, thousands of rounds a year shooting skeet, some trap and sporting clays, plus pheasant hunting and defensive training.
At a Gray Guns (now Op Spec Training) shotgun class we had clay pigeons on stands made for us by Karl at GT Targets. There were 8 targets in two staggered rows and you had to load with 5 rounds and run your gun dry, load 3 and finish the course. I was shooting my Mossy 590 and loading from a sidesaddle. I was the only Mossberg shooter, there were a few 870 shooters (who didn't come within 3 seconds of me) and a few people shot a defensive course with tricked-out Benelli's and 1100's, reloading from competition gear. It kind of defeats the purpose of a tactical class, but some people will do anything to be the top shooter in the class. Well, that didn't work for any of them because I could still run my completely stock 590 (except for the addition of a sidesaddle), hit every target, load the chamber, load two more into the mag tube and take out the last three pigeons at least 0.2 seconds, and usually more than even the 3-gun shooters.
Some instructors say a good pump shooter can outrun a gas operated semi auto. That I really don't know but I could beat the auto shooters with my Mossberg because I've trained constantly with all aspects of shooting my pumps, especially reloading from the sidesaddle. I have mine set up two shells facing up and four down, and drill with dummy rounds for hours. You get out of good training what you put into it.