This sounds like a classic case of fuel starvation.

I'd say it's got a blockage in the fuel supply tube that picks up the fuel from the tank up into the carby or as Norm suggests a bit of gunk that's being pulled up and gradually block the supply. After it stops and sits for a while the blockage reseeds downwards and again the process is repeated after it's been restarted.

I'd follow Norm's advice and see if that solves the issue, if not then its a carby off and blown out all holes with compressed air.

Cheers,
BB.

I live a 24 Hour lifestyle, but every now and again I seem to fall asleep, well at least that's what my wife tells me.