The use of pumice and rotten stone ( Tripoli ) I’ve done is not with a brush, but with felt pads and fine pounce pads like those used in French polishing
There are a couple schools of thought on the exact process,depending on the sheen you are shooting for, as with any skill. But generally the pumice does the coarse abrading between coats and is followed by the finer Tripoli at the end if you are shooting for a higher sheen. I’m certainly no expert with it, and have only gone through the process when forced to.
They are rubbed in with the pads and either small quantities water or oils like raw linseed oil or mineral oil, then cleaned off with naphta or mineral spirits. Water is not compatible with a shellac finish. I just use mineral oil from the pharmacy.
I have read that residue on carvings is cleaned off with a shoeshine brush, but I have no first hand experience with that. A fine brush might help dust off the residue.
The first piece I ever made at age 9 was polished off with rotten stone at the behest of my mentor. I remember getting sent back to try some more several times until the piece passed muster. I remember the process took me longer than building the piece, which explains my aversion to it.