I always remove the cylinder and let it soak in a jar of Hoppe's No. 9 while cleaning the barrel and frame. As needed, I use a bronze-bristled brush inside the window and especially around the forcing cone. I always go over the entire frame with a used tooth brush and plenty of solvent. By the time all that's clean, the crud in and around the cylinder is usually loose. I would never use a stainless brush on a blued gun; for cleaning cylinder bores, however, I've found it helps to move up one caliber, which is to say: I use a .40 caliber bronze-bristled brush to clean the cylinder bores of a .38 special or .357 magnum. Again, I use a bronze-bristled brush on the face of the cylinder and extractor, and a used tooth brush everywhere else.
Cleaning up a used, neglected revolver is usually easier after it's been stripped to the frame and left to soak in mineral spirits a day or so.
I never shoot .38 special loads in .357 magnum revolvers: too messy to clean afterward. I load very light .38 special loads in .357 magnum cases to make cleaning cylinder bores easier.