It is possible to isolate a small number of springtails from an infested culture to start a new one. You basically need to capture individuals and relocate them to a new setup. One method is to place a food source in a petri dish in the culture for a short period of time. As soon as you get springtails coming to the food, remove it to the new enclosure and let the springtails hop off without putting the dish in. If done correctly, you won't bring any mites with you.
For larger feeder insects such as roaches and crickets, a cornstarch bath works great to remove mites from the insects themselves. Simply put cornstarch in a tub, place the insects in, close and gently shake for a minute. The cornstarch makes the surface of the insect slick so the mites can not stay on. It's not perfect, and you'll likely transfer some mites over, but if done systematically, you can kill a mite infestation with a few periodic baths. Talcum powder works too but I prefer the cornstarch.