One question came to mind though, if your fire forming a cartridge like an AI would you get better results if you slightly lube the case so that your moving all the brass in forward motion.
If the parent case that is about to be AI-improved by fireforming in the rifle chamber is headspaced properly (and they are supposed to be pretty close, by definition of an AI cartridge), then adding a light lubricant coating to the casing could only help, but likely wouldn't be of great benefit.
The benefit would be proportional to the excess headspace present at the time of fireforming. Using a somewhat reduced pressure load might also reduce potential for case web stretch, however the pressure needs to be high enough to form the case to the chamber or the fireforming effort was nothing more than wasted time.
I typically Flitz polish my chambers in the effort to enable easier case extraction. It doesn't take very long at all,
and it serves to reduce the friction coefficient of the chamber walls. I find a bronze brush for a .45 caliber or there-abouts, and wrap a layer or two of a piece of rag around the brush which will cause firm contact between the rag and the chamber. Screw a section of cleaning rod on to the bronze brush. Then chuck the other end of the cleaning rod into a battery operated variable speed drill. Coat the rag with some Flitz and work the Flitz coated rag in and out of the chamber while power turning with the drill. Flitz is safe (impossible to damage the chamber surface) and will buff the chamber walls to a finer, friction reducing finish. Clean up the chamber and bore when done and add a light coating of corrosion protecting oil to the chamber. Go shoot rifle with a nicely polished chamber, which is moving the friction coefficient lower. I don't Flitz polish to reduce case web stretch and separations. I polish so that if I push pressures higher than I should have, the fired case will be just a smidge easier to extract from the chamber.
I don't expect to be able to fire cases with excessive headspace sloppiness and get away with it by adding some lubricant to the case/chamber interface though.
The case head separation and case web stretch with the case in my photos largely occurred during my fireforming process. I used Lapua 30-06 casings to create 280 RCBS Imp cases. I ran the -06 cases into my 280 RCBS full length sizer to neck down to accept the .284 bullets. The 30-06 brass would not correctly headspace in my chamber. So I seated a 175 grain .284 caliber bullet in the cases and left them seated long - jammed into the rilfing. I was depending on the bullet jam to hold headspace properly during the fire-forming. Well a couple of the cases pushed forward in the chamber when the firing pin struck the primer - setting the case head forward and away from the bolt face. Then as case pressures rose, the case sidewalls grabbed the chamber walls, and finally the case head was driven back into the bolt face. At that time, the case web stretched and began the process of a case head separation. I knew I had a few of these in my batch of fireformed cases so I was using them while developing and identifying maximum charge loads for this rifle. They lasted several more firings with the proper headspacing. The one that separated would have been good for another shot or two if I hadn't screwed up and shoved the shoulder back the extra 0.008".
ANYHOW, I will lightly lube the case exteriors whenever fire-forming brass in the future. It can't possibly hurt. It could only be of some benefit. Measureable benefit? Maybe - maybe not. As I type this now, I'm wishing I had been lubing cases while fireforming brass for my .280 RCBS Improved. lightbulb
Lubing takes so little time and expense. So what's to lose? That's why I'll lube during any future fireforming. BUT the most important step in the process is to use a method that ensures firm headspacing contact for the non-belted cartridge prior to fire-forming. Lubing is no substitute for maintaining firm headspace contact during fireforming operations.