My soft doors also fall victim to permanent mud stains...I've tried various brushes with detergent but nothing really seems to work. The special soft top cleaner they sell might work but I don't really wanna spend $10+ on it.
After the hard winter I had mud and such on my soft top that I could not remove with normal washing. I used BestTop cleaner with a soft brush and it cleaned everything. Just be sure and not get any on the windows. Most of my dirt was on the back of the top and I tucked a trash bag up under the top to protect the back window and it worked fine.