Mites come from anywhere and everywhere they can. Mostly they'd come from wild birds, the grass they've sat on, the tree they've sat on, placed they've landed, etc. I am not familiar with the variety of mite that stays on mice, but I do know that most species of mite are particular about which animal they like to cling to/suck blood from. Some are ok with transferring to another species, some arn't.
Your best bet isn't to worry about where they are coming from, but rather to take preventative measures. A good way of doing this is doing a weekly dusting of your birds and their bedding with either Diatamaceous Earth (food grade), or Seven. It's like your approach to keeping your house cockroach free: an ounce of prevention is worth a pound of cure.