I would suspect the fuel shut off switch.
On a Diesel engine the only way to turn it off is to shut off either the air or the fuel. Normally there is a fuel shut off solenoid that stops the fuel when you turn off the key. I would check to see that when you turn on the key it is opening the fuel.
I'm not familiar with you Kioti, but I have had this issue driving truck in the past. On a steep grade, pulling doubles and the truck just died. I could start it and it would run just long enough to get me about 20' at a time with an occasional near wheelie in the tractor. Long story short, I did this enough times to get off the road, turned out to be the fuel shut off switch had failed.