dawn does FOR CERTAIN (it's what i used before applying FF). i don't know about joy or palmolive, etc., but i do know the surfectants and detergents used in these soaps are designed to cut through food and grease, so it stands to reason that they're pretty harsh. i would guess they have very high levels of sodium (or ammonium) lauryl/laureth sulfate, and that's harsh. SLS/ALS is an industrial cleanser.
for regular washing (not stripping), find a gentle brand and stick with it. i know FF makes one, and there are many others at local stores.