Loading for PRS, I often had to deal with cases that had been on the ground and were just nasty. Often the cases were stepped on, and almost all the round would have "D-shaped" case mouths. I would also have to process a lot of cases very quickly. I opted for a progressive press for doing this. I used a process that was contrary to conventional wisdom, and similar to OP's scenario.
First, I would tumble the brass to remove debris.
I would run the brass through the press twice. First time through, I would use a universal decapping die in position one, a mandrel in position two to get the necks round again, a body die in position three, then an S-type neck die in position four. Then I would tumble again so that cases weren't sticky from lube.
On the second time through the press, I would run through the universal decapping die to ensure the flash hole was clear, prime on the downstroke, through a mandrel again, then to a funnel die, and then seat.
This resulted in two pulls of the handle per piece of brass. I would usually get 0.004" of TIR or LESS, often LESS. I had tried the same process with a regular FL die with an expander ball initially and results were all over the place. Was it the press? Was it the die? I don't know, but I changed things one process at a time until I got desired results. That is where I ended up. I think its important to recognize that processes may have different results for different people. There is no universal way of doing things, only trends. As always, there are multiple ways to skin a cat. Its the results that matter most.