Glad to see someone else understands what an oven should be used for. Occasionaly my wife finds car parts in the freezer and her electric skillet in the garage (I use it for warming parts).

I don't leave the filters in the dishwasher for the heated dry cycle even though I expect the under hood temps are much higher than in the dishwasher.

I made the mistake of using soap once when I cleaned them in the dishwasher and the dishwasher filled to flowing out the vent with suds. In fact it had so much suds it couldn't drain.

I use dishwashing soap. The purists will tell you NEVER due this. My filters come out mostly white but with some red stain showing on the edges of the cotton. Washing out by hand is recommended and the filters will always remain black. Visually I see no difference in the filter element after my dishwasher and a new filter. I don't know what a lab test would show. I would think that water would be less abrasive than a brush. I've been doing this for over ten years in the same engines with as many as 350,000 miles on the engines.