My sixpennyworth, I use an orange nyweb with the grain, at 600 grit it gets into the grain and gives it a good clean out. can't see that cleaning between the grits will do any good, but it can't do any harm,
I only use.one grit of paper on a pen intended for a ca finish, there is little point in sanding above 240 if you are going to coat it with glue, but if using friction polish, I take it to 400, exceptionally 600. The bushes get coated with a thin layer of glue anyway so this benefits in that the bushes are not exposed to paper regularity anyway. The bushes are guidelines only and normally you shouldn't be turning absolutely flush anyway. Sanding doesn't have to be a forceful test of strength.
My point above, I think most people sand too much!
I used to get discolouring when turning light walnut or zebrano as the polish was cleaning the bushes and dumping the crap on the wood. Probably did it on other woods but it didn't show. Swapped to a spray sanding sealer and problem went away.