In the 130+ years of machine gun use there have been several methods of cooling machine gun barrels:
Early on water jackets were used. This was effective but heavy.
The French Hotchkiss machine gun used large cooling rings around the barrel.
The Lewis gun had an aluminum barrel shroud. The design was supposed to pull air across the barrel to increase cooling.
Heavy barrels were used.
How effective the last three methods were is debatable. None of these methods seems to have much use today.
The multi barrel guns are not really machine guns since they use external power so are not part of this discussion.
Was the barrel shroud on the U.S. M1919 series supposed to aid in cooling?
Modern machine guns tend to emphasize weight reduction. But for vehicle mounted guns this is not so important. They could have any of these systems added if they were truly worth while.