Maybe Neil Antin in another update, can show methods to keep Cleaning Bristles in their cleanest condition.
I gave up using any dry brush on a record over a year ago. I just use the Teflon rod that I address in the book with current updates. For me it works good enough and is easy to keep clean.
To clean a dry brush using a dry process you may want to try a low-tack silicone roller - Amazon.com: YYQTGG Dust Removal Roller Plastic Handle Electrostatic Safe Low Tack Silicone Film Roller for Fiber Removal Circuit Board (10in) : Tools & Home Improvement. I use the 4-in version to clean the platter mat that I use. It does not appear to leave any residue and they last a long time. The roller is easy to clean under flowing water (use your fingers to gently clean) and then rinse with DIW, shake and let dry.
Take care,
Neil