We have a natural spring that runs behind the cottage. This area is constantly muddy year round and the dogs think its a hoot to run through it. And drink from it! Yuck!
To answer your question....after they've been running through it, I leave them outside as long as possible. I've found that the mud just dries up and falls off. I don't need to hose them down. Their paws are sparkly white again in about 1/2 hour.
If for some reason I need to bring them inside earlier, then I just towel them. As long as their feet are dry, albeit dirty, they won't leave mud tracks all over the house, though you might have to do some sweeping after.
My tuppence worth