Check the primer seater adjustment. If it gets to high the shell can hang up.
Take a look at the rod that moves the shell advance. Make sure the pivot point on the bottom plate is allowing the rod to move freely back to the position where it can engage the plate to turn it. Powder and or shot can get caught and impede free movement of the counterweighted arm.
Check the shot drop bushing area for a flattened trapped BB that might be impeding the plastic bushing carrier from moving. It should be moving freely when you have a shell in the station or you can pull the operating arm slightly to allow the plastic bushing carrier to be moved by hand. If you do this, BE SURE TO HAVE SOMETHING IN PLACE TO CATCH THE SHOT. I run into this problem every so often when using the smaller size shot. I have also replaced the plastic carrier once when it got too scored to keep working. Also check the rubber seal in this area.
Check the pivot points on the rear arm that holds the gas assist until the plate is at the bottom of its travel. If the latch releases early or does not catch on the up stroke, the arm can catch the shell plate and jam the works. It should only release when the bottom flat plate hits the release arm to free the gas assist.
There is also a chance that you may have a BB or two trapped under the shell plate.
Word of warning, don't try to force anything on these machines. It WILL break, and replacement parts seem to be made of unobtanium. When you get the shell plate turning freely, don't be surprised if the gas cylinder still cannot advance the shellplate, as it has most likely lost much of its pressure over time. I have 2 of the 3.1 machines, a 28ga that is now a paperweight, and a 12ga that I use about 1 time a year. They were fine when they worked, but not really designed for turning out large quantities of shells.
I used to get "binding" when the large nut under the powder carrier came loose - make sure its tightened up. Unless you really need the auto gas assist, I might suggest contacting Hornady to get the couple parts allowing the machine to auto index, but without the cylinder assembly.
For more help, consider contacting me off tread via PM or email.