I have a different process for consideration.
I have this grill brush with a sharp-ish flat metal side with a bit of curvey point edge on the sides. Anyway I use the flat edge to get the bigger stuff off, the side to get into the twisty bits, and then brush with the strand direction. Then I put them in the washer and dryer. I never wipe or try to clean anything while wet or damp, because it seems to just smear into the strands.
I'm mostly happy with the results. I can't be as rough with the sisal ones, but the cotton holds up well to the wire brush abuse.