Neodymium magnets. I use circular ones that are small. A 3/8" diameter neodymium magnet, 1/8" thick connecting to an opposing magnet in the cabinet face, will hold all but the most heavy doors enduring a lot of jostling. I sometimes just use a magnet and a piece of steel or a steel screw to save on cost. With an application as described, I'd use four magnets, two in the door stiles and two in the cabinet face. I drill the correct diameter shallow hole with a Forstner bit and epoxy the magnets into place.
The magnets allow for a flush fitment and no worry about material getting into Velcro and stuff like that.