Our process is to first attempt to minimize how much gets dirty during any given meal. We collect all these items in a plastic dish tub.
When it is time to do the dishes any remaining solid food is scraped off or wiped up with a paper towel and put in the garbage or fire (if we have one).
Then we heat a small amount of water over the stove an add biodegradable camp soap. Once the water is almost too hot to touch we use a scrub brush to clean the dishes, rinse them using minimal water, and put them to dry on a net under our camp table. One key is to wash things in order of clean to dirty to keep the soapy water clean for as long as possible, so usually scraped plates (little residue) first and and anything with raw meat or eggs last.
Stainless steel can be tricky if it gets really dirty. I carry a non-stick pan for things like eggs, and we make sure to not burn items in the stainless pans so that they can be cleaned with a scraper and water. If anything did burn on we have Bar Keepers friend at home.
At the end of the season I run everything through our dishwasher.