Walking (or other activity) promotes rapid movement of lymph out of the tissues, by several times the normal resting rate. It is powered by the muscle system. Anecdotally: After walking, I sleep more deeply (in a manner that I think has to do with moving lymph out of the brain). I suspect that moving lymph to clean other tissues in the body helps the brain move more lymph when it sleeps.
Study on muscle action and lymph flow that indicates this is well established in multiple studies and indicates 3-6 x faster clearance with activity, though I have seen higher numbers elsewhere and, to my surprise, cannot readily locate that info at the momemt:
I am unaware of any papers on the piece I described as anecdotal and my opinion. Though a quick Google gets me this piece that claims it is by an MD and, among other things, talks about exercise and brain health:
