I think Dawn is a detergent, which is higher in salts than soap. It will still kill bugs on contact the same as soap, but if the plant is in any sun you should rinse the Dawn solution off the leaves after a few minutes. Should do the same with any soapy water solution, actually. Most dish soaps have some detergent, but Dawn is hyped as the "grease cutter".
http://www.quirkyscience.com/d...
I do get spider mites on both the upper and lower surfaces of the leaves of my Brugmansias, which are the most mite-prone plants I grow. But soap or detergent in a mild solution should kill whatever those little crawlers are, if you hit them with it.