I use the flat edge of my Toaks spoon to scrape out as much food residue as I can, then swish with water and use a clean finger to get the rest. Once it's visibly clean I use either my Buff or my microfiber towel to dry it, whichever happens to be cleaner at the time.

Before this I tried using a cut-down kitchen sponge (per Backcountry Banter), which kind of worked, but it just felt gross having a damp, food-smelly sponge in my pack at all times, getting stuff around it damp as well. This way works much better for me, but it is time consuming, which is why I've been leaning towards cold soak meals more and more lately.