Just my experience, but I can clean a barrel by running several patches of Hoppes No. 9 until I get a completely clean patch. I then run a dry patch down the barrel to remove any excess left over No. 9. I then run a patch with CLP on it and it will *always* come back DIRTY. Generally I can run about two or three more patches with CLP on them until I again get a clean patch. At this stage I run one final dry patch down the barrel to remove any excess CLP, but of course it does leave a *very* thin coating of CLP in the barrel. I view this as a good thing since it helps prevent any flash rust that might form in a completely oil free dry barrel.
The above experience tells me that the CLP is able to remove deposits that the Hoppes No. 9 can not. Don't know why, but I've gotten these consistent results on too many weapons to chalk it up to a fluke. This is now my standard cleaning procedure for the barrel of every gun I own.
I have heard people who clean their barrels with a lot of CLP on their patchs and *don't* finish with a dry patch complain that they get a few flyers on the first couple of rounds fired after a cleaning, but I have not experienced this using my technique of running a dry patch down the barrel as the final step. Presumably these flyers are caused by an excessively thick oil film left by the CLP, but as I have said I don't see this effect if I run one final dry patch as my last step.
I'd be curious if people that try this approach after a shooting session get the same result.