Most of the supply houses sell either a knappy fabric or mole skin to refurbish them. It is a real bear to get the old stuff out. I would up gobbing mine with aircraft stripper, which as far as I know eats everything but steel!
While restoring the window channel felt on the Fastback, I originally used the self-stick type offered by numerous Mustang vendors. I stored the relined channels away while they awaited the other parts of the door to be equally restored. I later discoverd that the self stick stuff was letting go and probably wouldn't hold up in the car. I removed this stuff and installed the original type of lining called "moleskin", which has to be glued in. This has been trouble free and is more similar to the original.