I would expect a fuel pressure drop ( internal fuel leak ) may be causing the problem. If you don't have a fuel pressure gauge to attach, the most common problem is the fuel accumulator. If you get the engine to running temperature, pinch off the small hose on the accumulator & let it sit for the amount of time needed for the problem to happen, then try to start the engine. If it starts up right away thats your problem. The accumulator is located near the fuel pumps.