We bag the old insulation in the attic, carry it out and haul it off, then clean the attic floor. That last step is the one people skip, and it's the reason the job is worth doing. New insulation laid over a dirty or damp floor carries the old problem forward, while a clean floor lets the new layer start fresh.
If the insulation looks like vermiculite, we leave it alone. We're not an asbestos abatement company, so we point you to one for that part instead of guessing. Once they've finished, we can come back for the rest of the attic. I'd rather tell you that up front than take on work we shouldn't.
Removal is usually step one. With the floor clean, we air seal, closing the gaps where house air leaks into the attic, and then put in new insulation. If the floors downstairs stay cold, the draft is often coming from below, and crawl space insulation is the next fix. Rooms that stay chilly even with a good attic may need wall insulation.