I don't typically put pipes through my dishwasher!Few hours; how many hours typcially?
But, as I always flush my pipes with very hot tap water, I can smoke them immediately after towel drying the chamber and pipe cleaner in the airway. Briar doesn't absorb much water at all. And the biggest stress it gets is when steam, hot smoke, and ember heat the briar, such as when you're actually smoking it.
Wood is more absorbent when it's very warm or mildly hot (over 200º is my guess).
BTW, I generally just air dry my pipes after the hot water flush, and let them dry overnight. Saves on paper towels and pipe cleaners, plus I am a bit lazy.