My guess is that the problem is that the engine is not getting enough fuel. It is OK when it's cold and warming up because it has the warm-running device enriching the mix, but once it finishes, the engine starts to run out of fuel. You can test this theory by disconnecting the linkage rod that runs to the pump so that you can move the pump's lever without moving the throttle plate. As the engine warms up and starts to stall, push the rod down so that the lever on the pump moves just a little bit to give the engine some more fuel without adding air. If my theory is correct, the engine RPMs will increase.
There is more information on the Linkage Tour sticky on the "Drive train, fuel ..." page.