A gas-powered leaf blower uses a combustion engine to generate power. Normally, when this engine burns gas, the vapors and heated gases are vented away, so the engine can stay cool enough to keep running. If the engine dies after warming up, it is likely the engine is overheating and automatically shutting down to prevent more serious damage.
Vapor lock occurs when the fuel already inside the carburetor overheats and the vapors cause the fuel pump to lock up. A vent on the gas cap allows those heated gases to escape, which prevents the fuel inside the tank and in the fuel system from overheating. To check for vapor lock, after the engine shuts down again, unscrew the gas cap slightly and try starting the engine again. If it runs without shutting off again, vapor lock is causing the problem. Replace the gas cap, or clean it thoroughly with a brush dipped in fresh gasoline.
Air Supply Blocked
As the engine starts moving, the internal temperatures also start rising. To keep the engine running, cooler air needs to be brought in, and the heated gases need to escape. If the air filter or muffler are blocked, the engine will overheat and shut off again. Take out the air filter and wash it in soapy water. Clean the spark arrestor screen inside the muffler, and scrub out the muffler and exhaust port with a brush. Brush off the fins around the cylinder.
The carburetor inside a leaf blower uses several plastic diaphragms to bring fuel into the carburetor, mix and measure the fuel and send it off to the cylinder. These diaphragms can warp after several seasons of use. This warping will cause the fuel to stop moving through the entire circuit, and the engine will shut off. Usually this happens when the engine starts heating up, as the diaphragms aren't completely shot yet, but are starting to go. Remove, disassemble and clean the carburetor. Check for any perforations in the diaphragm and gaskets.
If the engine is dying in rough fits and shakes, and the starter rope is hard to pull out after the engine dies, the likely cause of the problem is inside the cylinder. If air is entering the cylinder from a leak around the seals, the result is a loss of compression. This compression is required to keep the piston and crankcase moving, and, without it, the engine can't start. Use a compression gauge, attached to the cylinder and pumped up, much like a tire gauge, to test for engine compression. If the compression reading drops off rapidly, take the leaf blower to a mechanic.