I play my clarinet every day and when I’ve finished I remove the mouthpiece and hold it under cool running water for a few seconds before drying it with a soft cloth as thoroughly as possible. I don’t pull the cloth through but dry from each end of the mouth piece in turn.
Is this a good thing to do or am I likely to ruin the cork or damage the mouthpiece?
I’m sure that’s fine. If you have the option to let it air dry that may be the best. Personally, I do have a mouthpiece swab that I use. Sure, there’s an argument that it could change the shape of the chamber over time. But I bet the table changes regardless, and by the time the swab has any affect I’ll be ready for a new mouthpiece. Although I’ve had my clarinet mouthpiece for almost 20 years now (yikes!).
I have some vintage mouthpieces for my saxophones that probably date from the 1920s or 1930s and 1940s. They play just fine and I have no idea how well they were taken care of before I owned them.
IMO if you use some type of soft, smooth cloth you shouldn’t have anything to worry about. I currently been using the little clothes they sell for eye glasses. But I wouldn’t recommend using those mouthpiece brushes they sell with some of those care cleaning kits. Unless it’s really bad.
Running cool water through the mouthpiece shouldn’t hurt it. As for the cork it’s going to go bad eventually and they’re not that expensive to replace. Some water systems have chlorine in them. I don’t know if it will shorten the life of the corks or not. Either way a little cork grease should help prolong the life of the cork. I just put a smidgen of cork grease on the cork every time I practice.
I do the same but I use a pull through, both ways
Just don’t ever use rubbing alcohol or cloths impregnated with alcohol on a hard rubber mpc.
Or the ProTec reed case. 
Nothing happened when I whipped the outside of the reed case but it about messed up the inside.
My process for over 50 years-- Yes cold water, inside & out. No cloth at all. If the cork gets wet (most times it will), dry it immediately. Use cork grease more often on the mouthpiece cork than the others.
Use a non-wire brush (bristles) to periodically clean inside the mouthpiece. Make sure you put tape over the metal(?) top on the brush. There is no need to dry any part of the mouthpiece except of course the cork. I have used one mouthpiece mainly for 25 years and another one after that for another 25 years with no problems. Of course over years the corks do wear out.