Not familiar with FLITZ but real familar with metal polish since this is what I do for a living, part of anyway. Please keep in mind that polish by definition means abrasive. If your polish is slightly acidic or abrasive it will etch a little of the metal each and everytime you use it which should make your mop a little black. This doesn't necessarily mean you are taking a lot of metal away. We polish 0.002" material in which 0.0001" would ruin the part.
A good way to find out if it is acidic is to clean using the WWII corrosive ammo method. After cleaning, run a clean swab with houshold amonia through the barrel. If it turns blue it typically indicates the precesnce of acids. You should get a little black along with it. Dry the barrel out and then run the a clean mop, lube,etc.
Please keep in mind that metal composition varies not only from rifle to rifle but from smelt lot of the original metal to smelt lot. What works good on one metal may not even be recommended for another. Bore plating would also be a factor. Plating by definition is a thin sacraficial layer of metal that helpls with corrosion and adds some lubricating properties. Key words are thin and sacraficial.
I've always been intrested in a faster better way to clean a rifle, but even with all the stuff I have at my disposal, I stick with the tried and true.....and I have a number of 50 year plus rifles that can hit far better than I can aim.