In the mid-Atlantic, as well as other parts of the US, the past few weekend races have been bone dry dust bowls. This ongoing rain in the region will change that. Rain is forecasted for the next several days, including race days.
I'm doing 2 races with ~2 hours between. I anticipate the 1st race will result in my bike covered in mud/dirt, etc.

I'll bring a 5 gal bucket and some brushes and towels. I'm not sure of the water availability so I'll bring a gallon jug of water too. I also have a pit bike that can be used if the primary bike is too caked up. I'll pack the bike stand too for easier setup.

What are some tips/advice on cleaning/maintenance on the bike between races?

One of those pressurized sprayers works wonders. You can get a fine stream of water that can very precisely remove any mud. Be careful around the bearings.

i have one of those hoses that coils up really small with a high pressure nozzle on it. i also bring a couple pairs of pliers. if the race is at a school or other institution, usually all of the hose bib handles have been removed but, the pliers do the trick nicely. the secret is to find one way, way, way away from the eyes. if you don't, you'll be descended upon by the mongol horde. they get especially miffed when you un screw your hose and walk away. with that said, one of the best post race meals i've ever had was made for me by a women in a very large RV who totes her uber junior age racer kids around the country chasing UCI points. I hosed down their 4 bikes while she made us grilled ham & cheese sandwiches with a hint of spicy mustard, tomato soup (scratch not out of a can) and tater tots. Dessert was her home-made peanut butter cookies and hot cocoa (not the instant stuff). See what happens when you share.

I'll bring a 5 gal bucket and some brushes and towels. I'm not sure of the water availability so I'll bring a gallon jug of water too. I also have a pit bike that can be used if the primary bike is too caked up. I'll pack the bike stand too for easier setup.

A gallon won't do it. Think about buying a sprayer for garden chemicals. Those work pretty well and usually hold about 2-1/2 gallons.

Look for puddles. Use large puddles to get the bulk of the mud off. A small plastic container works well to scoop up puddle water and splash on caked up areas.

Have fun.

FWIW-most of the races I saw say that there will be washers available. I hope.

It's raining a good bit, and is supposed to keep raining till tomorrow. BUT it's been really, really dry for a really, really long time. The ground can soak up a lot of water right now without getting very soft. I rode last night at a local park and it was wet but still rock hard. No mud. I just walked my dogs at the park and it's just starting to soften up.

Anyway, Saturday races will be messy, but I think Sunday will be much less so.

I think you're right.

Decomissioned water fire extinguisher- 2 gallons, pressurize with a bike pump, look on Ebay

Changed my mind. Water is laying everywhere in the fields.