The slate being pourous will soak up a lot of moisture. I don't know that it will warp it or swell it or anything like that, unless it has a backing, but it may take a while in a dry environment to dry it out. Why are you asking about washing it?
I was yesturday setting up 9 foot Brunswick Metro and while I was on my coffee break the owner properly cleaned the surface. When I started leveling the slates I had some major issues with bended slates. The worst ever without obvious reason. I just thought if there is connection.
For 3 cushion billiards, don't they heat up the slate so they regulate cloth speed by controlling the moisture? Wouldn't the same principle work by putting an electric blanket on the slate to speed up the drying process?
From a material stand point, it is possible to warp. Due to slate being from sedimentary rock it is within the realm of warping. I have had other sedimentary rocks ( tiles ) warp as fast as overnight ( but rare ). I would be more concerned with efflorescence ( leaching of minerals ) than warping.
ive always damped wiped my slate before putting on new cloth. cleans off all the fine dust and any repair debris. never a problem. over the years of use i even damp wipe my cloth to remove chalk dust and house dust. it evaporates rapidly. just think of it as running an iron over silk. dont press down really hard, just enough to rub the cloth surface. as for the slate, doesnt matter how hard you press. thats my opinion having done this for nearly ten years without problems to slate or cloth