If this is polyurethane, and you do not want to use a stripper, which would be much easier, you can then sand the stuff down. This would be a very time consuming job.
I would like to remove the doors, take them outside and set on saw horses and strip them down. Much easier.
The alligatoring is a prep problem or too much applied and not allowing each coat to properly dry before the next coat is applied.
If the doors are not special, you might even want to procure new doors.
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