This old chestnut...
Well if dropped in water yep hang up under pergola by bridles (I have 4 lines on sliders so I can pump it up a bit and let it find its own size/distribution) and wash.
Obviously they need to be totally dry as explained, hence sometimes I use a small towel to get under that leading edge/canopy contact..plus it gives me another chance to inspect the kite.
If not, I always do a mid season and end of season wash same way. But really I only do this to carefully inspect the kite and plug any pinholes etc and it gives me an activity to do while I drink a beer or two and reflect how fckn lucky we are to have this sport and its associated technology.
Other than that I pack my kites in a extra large canvass bag, this way they can be rolled really loose and easily packed away. Theoretically avoids pinholes from sand abrasion in those stupid bags they provide...I'd rather buy kite without bag for this reason. But I wouldn't claim this pin hole saving technique is fact.
All lines, harness, wetty etc put in a big bucket of water after every session.
Then again I know crew who never wash their gear and seem to have no problems.
For me it is just peace of mind, that's all. I wouldn't claim to know what is best.
So my message is, do whatever fits right with you. If freedofrog wants to wash his kites after every session good on him. Me, I got kids to wash, feed and get to bed (and listen to my wifes daily events in exquisite detail

, after the sun has gone down, so washing my kite is not possible!