Apart from having to wait for the brown to use itself up, which takes about a week, I rather welcome the after-rust. It only makes the brown job deeper. Naturally, the steel requires carding off and re-oiling, but as long as you are not on a contract when it must be done by (pick - a - date), let it come to fruition.
But to answer the question: some browning solutions are worse than others for residual rusting. When the barrel, for example, is finished, I take it to the laundry sink, plugs still installed, and scrub it with a mud of baking soda and hot water using a stiff brush. While I am at this, I bring a full kettle of tap water to a boil, and slowly pour it lengthwise over the barrel, until the kettle is empty. this it's back to the shop, barrel in the V blocks, and rub it down with a piece of towel and MOTOR OIL. There are likely better oils for the job, but that's what I use. The barrel is still very hot from the flushing, so the steel really enjoys this oil bath. Then I immediately remove the plugs, and clean and oil the bore. Using this system, I get very little after-rust, and a re-appliation of the oil seems to take care of it.
Laurel Mt. Forge's browning solution is the most aggressive I have used.