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"Not a lot" is the answer where I'm concerned, but I try to swim where I can, and wear wool undies. Longer tours try to get a Warmshowers host every few days to wash and launder. But then since I stopped showering every day and wearing deodorant I don't get anywhere near so rank ¯\_(ツ)_/¯
Mud is the real problem, so I make sure cover up completely in wet weather and wear long johns at night.
https://www.decathlon.co.uk/p/swimming-microfibre-towel-size-s-39-x-55-cm-double-sided-blue-green/_/R-p-341947?mc=8751952&c=tropical%20blue_prussian%20blue_steel%20blue
One of these will also work. It looks too small but it's adequate. TvH's link looks better though.
I'm also morbidly curious as to what multiple night wild camp people do for hygiene but I think the answer is "not a lot" which is probably part of the fun for them. I'm sure nothing terrible happens so long as you can sink wash your base layers semi regularly.