When it dies, restart it and hop off and wait for it to die again. Once it stops try the primer pump and see what the lever pressure feels like.
When you move that lever you're pulling the diaphragm back against spring pressure, release it and the spring pumps the fuel out, until the fuel pressure builds and holds the diaphragm back so there is no pressure on the lever because the spring is held back also (or there is pressure on half the stroke because the spring only pushed some of the fuel out).
There is more pressure on the lever when the inlet is plugged and the lever/diaphragm is pulling against a vacuum.
You can tell if the pump is pumping air and not full of fuel because it will have normal pressure pushing the lever, but it will spring back with the same pressure, no fuel to dampen the spring pressure.
If the primer pump builds pressure in a couple of strokes then you can look downstream for a clog, or put a gauge near the injection pump and see what you get running and stalling.
If it's pulling a vacuum then look on the suction side, blow back into the tank, take the fittings apart individually, or pull the strainer out of the tank and clean it.