The problem you have sounds like a problem i had with my 1982 Honda ATC200 (i know, its not a moped...bear with me). The CDI module that builds up the charge for the spark was mounted under the fuel tank, over the head of the engine. Heat from the engine would heat the CDI up, and over time this would cause the CDI to fail. I would shut down the engine and have to wait about 15 minutes for the CDI to cool off before it would start back up. Someone I know has a Moby that I am looking at buying, and it looks simular, where the CDI is under the gas tank over the engine.
A way to test this is to run it until it heats up and than shut it down. If it refuses to start, disconnect the spark plug wire from the spark plug and plug in another spark plug, and lean it against the engine block, so it has a ground. Than crank the engine and look for a spark. If there is no spark, i'm pretty sure its your CDI that's bad, and you just have to replace that and it should start right up again. Worked for my 3 wheeler :)