Copper bristle brush, GM engine top cleaner, or a gun powder solvent. I am using GI bore cleaner because it does not eat up copper bristle brushes.
There is nothing mysterious about cleaning dies. They are metal, and what they are fouled with is wax, powder residue, and maybe some lead. Soap and water will remove the organic stuff, more aggressive solvents, like engine top cleaner or bore cleaners will remove the stuff faster. You will need a mechanical means, a brush or something to scrub out the packed in metals.
I do not recommend nasty chemicals like brake cleaner, carb cleaner. Not because they won't remove the organics, but because that stuff is nasty for humans. Those things will eat up your skin, probably your lung tissue, and once in the blood, who knows what the long term consequences are.
P.S. Wear nitrile or thicker gloves when using bore cleaners. You can get them from Harbor Freight Tools,