I'm thinking this is a carburetion problem. It sounds to me like the startup/cold running are okay because the chokes are covering up the problem with a richer mixture. Then, when operating temp is reached, something (a bimetal spring, maybe?) opens the chokes and the problem shows itself.
If the carbs haven't been cleaned and rebuilt, probably that's something that is needed. Can you do this yourself? It's not particularly easy, especially when the carburetors have wear or warpage of critical parts.