Well, I was the wardrobe mistress (manager) in the drama department when I was in college. We also rented out costumes to the public, in special circumstances.
We did some dry cleaning, laundering a lot (I am an expert at hand-washing large items!) and a LOT of steaming. We steamed the hats and wigs. I used lavender oil or tea tree oil in the steamer (expensive, professional model) on the synthetic wigs and they always look and smelled wonderfully clean. On the human hair wigs, we just washed them like you wash hair (gently, of course).
I would assume that costume shops dry clean the costumes. If you have concerns, I would call and ask them what they do. If they say they dry clean them, ask for the name of the dry cleaner (tell them you have a costume that you want to have cleaned). Call that cleaner and ask them about what they do (and, IF they do) for that costume shop.
We always insisted that cast members wear t-shirts under the costumes and/or dress shields, to help avoid perspiration contamination. Dry cleaning was/is expensive and toxic, and anything that can be done to avoid it is a good idea! Dry cleanable-only fabrics (or, vintage items) often had full cotton liners that we custom-made (like a chemise) worn underneath. We also made sure most of the collars on costumes could be easily removed from the garments for ease of cleaning, too, or we sewed cotton liners (that could be unstitched) to be cleaned or discarded.
Now, with products like Febreez on the market, I am sure that some costumers just use that stuff. Like I said, dry cleaning is expensive and, depending on the costume, some things require extra care (beading, feathers, materials).