Hi BF
Here comes misery guts with no banter.
We ran a riding school for four years and looked after 30-35 horses. Assuming the rugs are the heavey canvas waterproof type we did the following.
We had a fence panel between two posts to hang the rug over along the back seam this supports it while you work and stops it flapping about.. You want a pair of waterproof trousers and then a hose with a very stiff brush to scub the worst of the mud and c**p off. Plenty of water to rinse as you go. This will get 95% of the dirt off. If you use detergent based cleaner it will destroy the existing waterproofing. A pure soap based cleaner would be OK but we rarely bothered as it would not make that much difference. They take a while to dry out but I would think if you put a decent vac on they would dry quicker.The tack shop will have heavy duty reproofer which is similar to tent proof.
Its a mucky job like most horsey things.
I've plenty more tles but thats enough for now.
Trevor