I just wipe off the grit from the stones, no real need to wash it or anything before proceeding. And as for strops, I keep pasted strops in the honing den and only clean strops in the shave den, minimizes my desire to use the pastes.

I don't see anything wrong with extra cleaning, even if it is unnecessary (between two clean strops?), just don't mix abrasives and your good to go.