End result:

When you peel off the fabric, the surface behind it should look something like this:

Get a stiff brush and brush off all the loose foam/glue residue you can. Preferably an old brush as it will end up rather sticky!
Grab a cloth and some thinner and start rubbing the remaining glue residue off the door card surface. Now is probably a good time to remove the handle to allow easy access to every corner. Afterwards it should look more like this:

It will still have some residue left over, but as long as it isn't loose or sticky then it should be fine.
Place the fabric piece face down on a flat surface.

Start brushing all the foam residue off the back pf the fabric like you did with the door card surface.

Apply a good coating of spray glue to the back of the fabric piece. Be sure to get every corner covered, and leave the glue to become tacky for about a minute.

Offer up the fabric to the doorcard. Stick it down as central as you can, smoothing it sown from the middle. Leave the edges loose for now so we can tuck them in.

Use a plastic tool or flat-bladed screwdriver to carefully push the edge of the fabric back into the groove. Run a Stanley knife around the edge if need be to cut off any excess fabric or stringy bits, but be careful not to pierce the door card material. And after replacing the handle and having a cleanup, it should be done!

Quite a satisfying job overall, and took about 10 minutes per door. Really improves the interior looks!