No. The "wrap" is almost always aluminum foil, or a (disposable or easily cleaned) pan covered with foil.
You can divide barbecue pits into two sorts. Those that burn hardwood logs for heat and smoke are called "stick burners." Then there are the other sort named according to their preferred fuel (gassers, charcoal-burners, 'lectrics, etc.) which use some other fuel for heat, whether (duh) charcoal, gas, electric, or something else for heat, and which burn pellets, hardwood sawdust, or hardwood chunks for smoke.
Of course, there are other ways to name pits. By their shapes for instance, yielding names like offsets, pipes, cabinets, and so on.
For what it's worth, anything with a large enough fire box can be a stick-burner. Most stick-burners are (built in) brick pits or they're offsets. Any stick-burner can also be a charcoal burner. Of those offsets which are too small to burn sticks effectively, most are charcoal-burners -- as opposed to gas or electric.
Before people jump in with stories of how they went all stick in their Silver Smokers or little Brinmanns, let me add that you absolutely can go all stick in a small offset. However, in addition to requiring constant tending, they're also very susceptible to imparting tastes from wood (which, for lack of age, moisture, mold, pitch, etc., might have been less than an optimal).
As it happens, my offset, a Bar B Chef which was too small to have been a good stick-burner, was converted from charcoal to run a gas fired "Afterburner-H."
All of which gets us to the following: With the 'cues which aren't stick-burners, the pitmaster often stops making hardwood smoke somewhere around halfway through the process, as the meat has absorbed all of the smoke which it can profitably handle, anyway.
It's not only a matter of efficiency and economy, but avoids "over-smoking" as well.
Wrapping meat near the end of the cook will not cause it to be "under-smoked," if that's your concern.
Some cooks believe that it negatively impacts the "bark." Others believe that it has a tendency to over-tenderize the meat. Still others think that it's insufficiently orthodox. Indeed, I've heard barbecue celebrity purists say, "wrapping is braising and braising ain't barbecue."
The reality is if you're serious about barbecue competitions, you've simply got to wrap. Comp seems like a digression, and it definitely shouldn't be the be all and end all for home barbecuers. Let's just say that the crucible of competition settled any questions about the utility of wrapping once and for all.
Those cooks who wrap, but also want a dry bark, usually open the pack for the last hour or so (which would allow something around a 10F increase for a 10# butt, cooking at 225F), to allow drier heat to get to the surface.
In my opinion, that's unnecessary with pork belly. It should be finished and served moist -- along with the "braising liquid" from the wrapped packet or pan.
Does this sufficiently address your question?