I have this very same issue, and I found that Scrubbing Bubbles worked well. You just need something below to catch the filth that will run off. The bubbles penetrate the porous surface and draw out the soot.
If you can’t get scrubbing bubbles, then baking soda and vinegar would create the same kind of bubbling action but you’ll need the right proportions so as not to leave a baking soda residue.