Some car covers shed when they get older. Looks like dust, but it's actually the cover itself. A family member has one like this that still looks fine, but it's way up there in years. When you shake it fine fabric particles fly off it like dandruff on steroids. It's annoying, but it blows off when you drive the car.

If you have a cover that you are sure is letting dust thru then put thin plastic sheeting over top of it.