Can be a lot of things.
Try checking for spark at plug right after it dies. If no spark maybe it's the coil or ignition module.
Is water coming from the tell tale? If not, broken impeller or faulty thermostat?
Maybe it picks up debris in the fuel system after some minutes driving. Try replacing the inline fuel filter and/or cleaning the carb(s).
Does primer bulb go soft or collapse when driving? If so something is restricting fuel.
Fuel tank vent defective? When engine starts dying, open the tank lid and see if it gets power again. If the engine dies very sudden, then just try having the tank lid open from the start.