The advice I got when we had Pasty Butt with my baby chicks was to hold a washcloth soaked in warm water up to the area to soften the poo, or to "dip" that end of the chicken in a bucket of warm water for the same purpose.
It wasn't quite enough to get the job done for my chicks so I had to manually extract the hardened stuff.
I don't know about prolapse, though, so maybe you need to do something different with that...
Maybe a little soak with some epsom salts would help sooth the prolapsed vent and soften/loosen the poo so it could be more easily washed off? I used Johnson's Baby Bath for pasty butt and it worked well for us.