If I cook something that is messy, which is rare, but not never, I'll bring some Dawn dish soap and if I'm by a creek I'll get some water and sand from it and use it with a few drops of Dawn to scrub a greasy pot, or I'll use a small piece of a Scotch scrub pad.

Dishwater disposal is the same wherever you go. I just walk a few hundred feet away from the source and camp and dig a cat hole and cover it when the water is mostly soaked up. I don't use much water though and I bring a thin wash cloth to wipe down my cup and pot if they need it.