I have no trouble at all cleaning cast iron skillets as long as I put it under some water while it's still hot and give it a little wipe. Doesn't seem to affect the seasoning either ..
Putting a bit of water (or other liquid) in hot pans is a generally good idea to turn your fond into a sauce anyways - it's a technique called de-glazing[1].
You should avoid putting some cookwear directly into cool water though - the temperature shock can cause cracking and other damage if it's coated. I don't have a great rule of thumb for this, you should just double check any advice you can find specific to your cookwear.
that's the key, take it out immediately using HOT water from the tap to clean it out works great. Even when cold though it's not that hard to clean with a plastic or wood scraper if it's properly seasoned.
You should avoid putting some cookwear directly into cool water though - the temperature shock can cause cracking and other damage if it's coated. I don't have a great rule of thumb for this, you should just double check any advice you can find specific to your cookwear.
1. https://www.bonappetit.com/story/what-does-deglaze-mean