Other than using a stiff brush (non-wire) and water/detergent/etc, cleaning ALL the mud/dirt may be futile.
I have used a product called Fluid-Film, a lanolin (sheep-fat) base product that acts as a barrier film. It works best if applied annually for a couple years, then touched-up as needed... mostly as an anti-rust treatment. Surprising, it was developed in the mid-1940's. Yes, its gel-like film can attract and hold dirt, but it can be washed-off easily, and creeps into minute spaces, hopefully to keep further dirt and corrosion from occurring. And, it works best if applied before the underside gets crapped-up.