It also looks to me like the case mouth (that is normally what the distal end of the neck is called to keep it distinct from the case rim at the other end) ran into the edge of the funnel/drop tube/operating rod rather than letting it enter the neck. That would be due to the case tipping on its way up to that tube. If these cases are being or have been fired in a self-loader, look for rims that have been bent outward by the extractor. You can find this by lining the cases up on a table top and sighting down them. If some of the mouths don't line up, they are tilting due to the bent rim problem.
Things that can help:
Pull the shell plate off and clean it and the press ram platform and reinstall the plate, paying particular attention to the last paragraph on page 7 of the instructions. Try to tighten the plate so as to minimize wobble and play as Scattershot suggested. It may help to burnish the bottom of the plate with some graphite or to apply a dry Teflon lube.
The brass buttons that keep the cases from flying out of the shell holder have to be sized to accommodate a maximum diameter case head and rim. Most cases are not maximum, though, and you can usually put a layer of tape around the head of the button on the #2 station to cut down slop and help center the cases more consistently. I actually turned several spare buttons that are oversized in half thousandth steps, but if you don't have a lathe, Dillon makes a .32 ACP conversion, so they have at least on button bigger than the one for the .223. You could get one and turn it down with a flat file while spinng the button in a drill until it just removes the slop in the orignial. You don't want it to squeeze, but only slight clearance could help.
Finally, just go slow until the drop tube is in the case mouth. Speed kills, here. You want to find where in the handle stroke the drop tube starts into the case mouth and get in the habit of pulling the handle so you slow down at that spot and can feel a faulty start and reverse the handle slightly and wiggle the case into position. Once the tube starts into the mouth, the rest of the stroke can be as fast as normal. The extra caution will only cost you about a second per round, so you'll not give up much of the 550's speed.
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