Backcountry water sources should not be used for dishwater disposal or soapy bathing. This goes triple for desert watering holes, where soap or food contamination can have dire consequences for aquatic life and the next drinker. Instead, clean yourself and dump your dishwater 200 feet from water. Dig a cathole, strain out food particulates, pour the water into the hole, cover, then return the area to its natural state. Eat, burn, or carry out food chunks so critters don't acquire a taste for human cuisine.
Here's how to Leave No Trace when you dump water in camp.