I had an issue like this, it turned out to be the heat shield on the bottom of the gas tank. Most have the fuel pump in the gas tank and with the tank sitting in close proximity to the engine, when the engine heats it "boils" the gas and the fuel pumps don't like this situation. I replaced the heat shield on the bottom of the tank and have not had an issue since.
Sounds like it could be the ignition coil. Usually if the bike will start and idle good but quite when its put under load (driven) the coil is the problem. when the coil heats up it breaks down ,once the coil cools down the bike will run until the coil heats up again.
It sounds like you may possibly have several problems. How old is the engine coolant? How old is the engine oil? I ask these questions only because coolant does loose it's ability to cool the engine after it gets so old. Also, if the engine oil is old or of the wrong type, this too will cause the engine to run too hot and when that happens the engine has a temperature sensor that will shut that engine down and won't start until it cools again. So if it's a cooling issue then you need to rectify that. Change the coolant 50/50 and change the engine oil. Use 10w40 synthetic oil and see if that helps. If it doesn't then you may have an electrical issue such as weak ignition coil, weak stator coil, bad spark plug as any of them will cause the same symptoms. In regards to the mechanics of the engine i don't think you have any problems there. However, most owners neglect to have the valves adjusted and that is a detrimental mistake. That engine has to be able to suck and blow on time in order for it to perform like it should. If it's late getting it's air that will cause the engine to run too lean if it's too late getting it's air then that will cause temps to go too high as well. If your not mechanically inclined my suggestion to you is take it to a dealer and have the issues checked out further.
Answers & Comments
I had an issue like this, it turned out to be the heat shield on the bottom of the gas tank. Most have the fuel pump in the gas tank and with the tank sitting in close proximity to the engine, when the engine heats it "boils" the gas and the fuel pumps don't like this situation. I replaced the heat shield on the bottom of the tank and have not had an issue since.
Sounds like it could be the ignition coil. Usually if the bike will start and idle good but quite when its put under load (driven) the coil is the problem. when the coil heats up it breaks down ,once the coil cools down the bike will run until the coil heats up again.
It sounds like you may possibly have several problems. How old is the engine coolant? How old is the engine oil? I ask these questions only because coolant does loose it's ability to cool the engine after it gets so old. Also, if the engine oil is old or of the wrong type, this too will cause the engine to run too hot and when that happens the engine has a temperature sensor that will shut that engine down and won't start until it cools again. So if it's a cooling issue then you need to rectify that. Change the coolant 50/50 and change the engine oil. Use 10w40 synthetic oil and see if that helps. If it doesn't then you may have an electrical issue such as weak ignition coil, weak stator coil, bad spark plug as any of them will cause the same symptoms. In regards to the mechanics of the engine i don't think you have any problems there. However, most owners neglect to have the valves adjusted and that is a detrimental mistake. That engine has to be able to suck and blow on time in order for it to perform like it should. If it's late getting it's air that will cause the engine to run too lean if it's too late getting it's air then that will cause temps to go too high as well. If your not mechanically inclined my suggestion to you is take it to a dealer and have the issues checked out further.
It sounds like the external coil is bad.