Penetrating oil, "Free All" can help loosen up the carbon, especially if you push a couple of patches though the barrel when you are done shooting. Free All is a 21st century penetrating oil...like no other.
Try some Montana Extreme copper cream, then some Flitz bore cleaner, both of which are easier to use than JB.
One main issue I have had over the years, that I recently discovered, was using the bronze bristle brush too long. Figure 50 strokes, put on a new brush.. Also, the use of FINE bronze wool wrapped in that brush with 50 strokes on it, cleans the carbon out very, very well. The Fine bronze wool does not scratch my fine SS Match grade barrels as witnessed in the Hawkeye and Teslong bore scopes! I can not stress this enough!
I was having one heck of a time with R#15, 8208, Varget in my 308 Krieger, it was taking around 100 rounds where the accuracy Where accuracy was ruined, from a 1/4" group barrel.
When you are seeing that Grey carbon start to accumulate, suspect your brush first thing. Try an experiment, mic a new brush, re measure it every 15 strokes, get ready for a shock. Those bristles have to have "spring" in them to scrub the bore.
I use the Copper cream and Flitz bore cleaner on patches, two or three patches, short stroking usually does the job.
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When you get down to carbon starting to accumulate, time to try a new technique/chemicals.