The riser inside a bell or barrel does not depend at all on heat coming in from the bell; it should be insulated and preferably lightweight so that the immediate heat of combustion warms the inner surface very quickly, ideally coming up to full operating temperature within 5 to 10 minutes. Heat will slowly pass through the riser wall, and will warm the bell some but hopefully not much beyond the hot gases passing down around it.
If the downdraft tube is similar to the riser, hot gases will move quickly through it. It will all get very hot, since the gases will have moved on before transferring much of their heat to the tube. There will be more heat left for the following duct or channel.
A bell, being very large relative to the riser, will let the gases move slowly down through it, giving up heat to the much larger internal surface, and leaving less for the following channels. This is the same for a barrel or a masonry bell, the difference being that the barrel lets the heat pass quickly on to the room while the bell passes the heat through layers of masonry on its way. Gases do not move as a plug leaving the perimeter undisturbed; they generally move as a whole, with some degree of turbulent mixing depending on the design.
I would never build a masonry bell without including an access panel for inspection, cleaning, and if necessary, riser repair or maintenance. This panel if sized and located appropriately would serve some instant heating function.