Our water is from a well - quite high mineral content - if allowed to air dry on about anything, it will leave spots - so I use water that has gone through our water softener as my rinsing water for brass. I was always concerned about liquid water ending up pooled in a case on it's side or on the case head - so usually I place them mouth down - thinking that allows them to drain completely.
As above - I do not use Ultra Sound for brass cleaning - but I use stainless steel pins in wet tumbler to clean and shiny up fired brass. Also, as above post, the fired primers are punched out before the cases get tumbled. Recipe for the fluid is as found on CGN elsewhere - pretty much healthy shot of Dawn dishwashing soap and 1/4 teaspoon Lemi-Shine crystals - likely will vary with the water quality that you have to work with - generally, if "suds" are left after the thing times out, that was too much soap.
I have stacked rifle cases mouth down in metal bread (loaf) pans and just left them "air dry" out in garage over night. I have fired up the oven to perhaps 250 F and put in a metal cookie sheet of "wet" brass for perhaps an hour. Water turns to steam circa 212 F - so just get the brass to be hotter than that and will be no more liquid water - but also is not about annealing the brass, so do not want to get close to that temp. Our oven has a fan - I presume that is also a vent, but I do not know, so "just because", I tend to turn on that fan when I am drying brass in the oven. Often let it go for an hour - then shut off and let it sit in there overnight - deal with it in the morning.
We can get Dawn dishwashing soap at local grocery store. Lemi-Shine crystals came from Canadian Tire. I bought a jug of those crystals several years ago - still working on that one - I have not had to buy replacement yet. I use RCBS Case Lube-2 to lube the rifle cases to re-size, AFTER tumbling them to clean and dry them. So, I use a Lee Case spinner - each case gets trimmed to length, chamfered and then spun against a wet rag to remove the case lube after re-sizing. Others, I understand, will use a tumbler to remove the case lube - then re-dry a second time.
I do have an Ultra Sound machine, but have been using that mostly for parts cleaning - 3 litres water and healthy shot of Dawn dishwashing soap - or 3 litres of water and parts in a plastic baggy or peanut butter jar with various solvents like Ed's Red, kerosene, Varsol or ATF - the dirt stays in the container - water in the tank stays clean and can be re-used multiple times. As I have previously posted on CGN, you probably want to distinguish between making your brass to be "clean" - no grit; versus to be "shiny" - which it appears that some equate with "better".