Have always had good results with Hoppes #9. Run 2 or 3 wet patches and let sit for 30min and then run another couple. Final patch a couple of times with your favorite gun oil. Like our BP barrels, smokeless are not all the same. Some take more or less work than others.

Chuck I have had real good success with using JB Bore Paste over the past 23+ years. 
If you are wanting to have patches come out perfectly white follow it up with some patches using Naphtha. The problem with doing so is it can also remove the prior barrels lube seasoning. Unfortunately it has now been outlawed in Ca. and no longer available here. It was also what Ronson and other Lighter fluids were actually made of. In your area look on the back of a bottle of Lighter Fluid to see if it still contains it if you would like to try it. Used to be I could go to the local hardware store and get a gallon for around 9 dollars. I did not know it was allowed until I went to go get another 1 gallon can when I was just about out. BBQ lighter fluid is also not the same. After the barrel is clean I just use a couple of patches using Ballistol. 

Lighter fluid or lighter fuel may refer to: ... Naphtha, a volatile flammable liquid hydrocarbon mixture used in wick-type lighters and burners.

Charcoal lighter fluid is generally "mineral spirits" not naptha
Coleman fuel is "white gas" not naptha
beltfed/arnie

+1 for Gunzilla.

For cleaning I am also a fan of Ed's Red. For plain based lead bullets with smokeless I use a patch wrapped bronze brush with Ed's Red 10 strokes and change patches. I usually go through 3 patches and I'm finished I leave the barrel wet.
My attitude is that if your getting your barrel squeaky clean (cleaning down to bare metal) you are over cleaning your barrel. Lead bullet and jacketed bullet barrels are two different animals. jacketed bullets ride down a naked barrel and tear off little bits of metal as they pass through. When cleaning you have to scrub and resolve the copper residue to get accuracy. 
In lead bullet barrels the bullet rides on a thin film of lubricant not bare metal. When you clean back to bear metal you are starting from scratch every time. Clean it until you get the carbon deposits and powder fouling out. Then your patch shows just cooked lube stop. let that lube get ironed into the barrel. This is what's meant by "seasoning" a barrel.
The bottom line is a lead bullet barrel has more in common with a rimfire barrel than it does a jacketed bullet barrel.

40 Rod

CRC Brakleen, automotive brake parts cleaner. Don’t get this on your stock finish, and don’t use it on newer guns with plastic parts.

Primers probably leave more abrasives in the barrel than any powder might do. When I clean I do 10 strokes in the first third of the barrel, 5 strokes in the second third and two strokes in the last third. This will eventually create a choke towards the muzzle over time. I also use a Parker Hale wrap Jag and I never let it exit the bore. I also have never used a brush but just wiped it until is clean. My 32-40 barrel as most of you know has over 140,000 bullets down the bore. At over a 100,000 it was still able to set a new ASSRA National 100yd. 5-5 shot Group average record back in 2013. Not at all trying to say this is the best approach by any means but only that it has worked very well for me. Most barrels don't see that many bullets going down their bore. So I just wanted to share my own 23+ years of my own personal cleaning process while using JB. The throat is now pretty much gone so instead of setting it back and re-chambering it I decided to just retire it so it will remain original to the rifle. As the troat continued to wear I just had to continue in seating the bullets abit more deeper.