Sounds like it could be a fuel flow problem. If the fuel filter is partially blocked, say with rust particles that have built up while the bike was sitting for a long time, then there would not be enough fuel flow volume to keep the carb bowls topped up while the engine is running. Then if it sits for a while, the bowls would slowly fill back up, the bike would start, but would only run for a short time before quitting because once again the fuel level in the bowls would fall too low.
One way to check this is to remove the fuel feed hose from the gas tank petcock. If the petcock is in the on position, no fuel should run out. If the petcock is then turned to the prime position, "PRI", fuel should flow easily (have something to catch in in). If it's only a trickle, the there's some blockage and the petcock will have to be removed and the fine-mesh screen filter cleaned.
If someone has added an in-line fuel filter, then the flow rate will have to be checked on the carburetor side of that filter.
May not be the problem, but it's relatively easy to check.
Keep us posted.
2000 Kawasaki Vulcan EN-500C, 20k