Michael,
The moly ring is a function of a poor moly-plating process. These processes typically produce bullets with clumps of moly on them or too heavy of a coating that does not adhere properly and that produces moly buildup in the throat area of the chamber.
A friend decided to try moly-coated bullets a number of years ago. He was using a process that did not work well. During his first HP Silhouette match shooting the moly-plated bullets he had serious pressure excursions that finally resulted in his bolt sticking badly. Once we got his bolt open he and I discussed his load. When his rounds were looked at the problem was at once obvious. His bullets had clumps of loose moly on them. Once the bullets were wiped off and his chamber throat brushed out to remove the moly build-up the pressure problems disappeared.
If ones bullets are properly moly-plated there is no moly build-up in the barrel. A proper plating process produces bullets that are shiny, with a thin coating of moly, no loose moly on the bullets whatsoever and the bullets can be handled with no moly coming off on the hands. Finally, the bullet meplats should NOT be flattened from the plating process. Flattened meplats cause substantially reduced ballistic coefficient of the bullet, not a good thing. Oh, and no wax should be used in the process.
As to cleaning the barrel when using PROPERLY moly-plated bullets, two wet patches with any good cleaning solvent like Butchs Bore Shine followed by a dry patch is all the bore needs. This simple cleaning process every couple of hundred rounds is sufficient so long as one is using high quality, clean-burning powder. If low quality powder is used, that deposits large amounts of fouling, then cleaning more often is required. I use the VVO N100 series powders. They are the cleanest burning powders Ive ever used. That certainly contributes to my rifles ability to maintain top accuracy with very infrequent barrel bore cleaning. Of course we are talking about match quality barrels, not factory barrels. The chamber and lugs should be cleaned more frequently with special attention focused on keeping the lugs clean and well lubed as well as the primary extractor surface on the bolt handle.