My indoor is dusty.
When I get off, I take a really soft face brush (almost as gentle as the horsehair finishing brush) and brush off my saddle, bridle, my boots/halfchaps as well as any boots that are on my horse.
Also, I'm really lazy and carry all of my tack, brushes, and helmet all at once.
I ride english, and keep my bridles wrapped (figure eight and wrapped noseband)
My stirrups are put up, and my girth goes over my saddle through the irons.
I like to stick my bridle through them as well, that way my arms are free for my grooming back and helmet.