I use both processes, mop and spritz. I spritz when I can, but thick liquids like EVOO doesn't spritz well, so those kinds of things I mop.
My inlaws got me a silicone mop that has wide flat strips. The strips have holes in them so you can get it loaded up. Not as efficient as a true rope mop, but very easy to clean.
Also, on my rope mops, I drill a hole in center axis of the handle and press a 1/4" bolt with the head cut off. When I'm done smokin for the day, I fill a bucket with soap water and spin the mop with a cordless drill. Cleans pretty easy, but hard on the mop....they don't last long this way. But when you get it clean, keep running the drill until the mop is mostly dry. I think this goes without saying, but make sure you are outside....and away from any object you don't mind being painted with q-sauce.