<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">The wrapping is done to let the meat rest and relax more than anything. </div></BLOCKQUOTE>True--if you are only wrapping after cooking. There are other reasons one might wrap during cooking as I noted above.
For ribs, I foil with a mix of pineapple and concentrated tamarind juices--single piece of HD foil sealed tightly. I do this after the ribs are deeply colored and then cook till done in the foil. Then I remove the ribs from the foil and allow the rub to firm a bit by returning the ribs to the cooker for a few minutes. If I am glazing I do so now, but I apply a thin veneer of glaze and it sets it a matter of a few minutes. I do not care for ribs that are sauced for serving so I never sauce ribs at the end of cooking.
Briskets I cook at high heat--either the mid-upper 200s measured at the grate--or, more usually, in the 310-320 range. I foil after the brisket breaks plateau (167-172 or whenever I happen to get to it), one piece of HD foil crimped tightly but with space around and above the brisket as they tend to throw off a lot of juices and I want room for those. I then take it to fork tender (temps can vary greatly when cooking briskets this way so I don't bother temping although I might leave the probe in to remind me to check it). I check the brisket at the just before the 4-hour mark (total cooking time) and it might be done or near done. I rarely have one go all that much over that time-wise. The way I mostly do briskets is described in
this thread. My philosophy on cooking temps for barbecue is explained
here.
Chucks I smoke 2-2.5 hours (I don't care what the internal is at that point) in the low 300s then foil, usually with a liquid reduction, onions, garlic and possibly other aromatics or vegs. Because of the additional ingredients I fashion a 'pan' out of two pieces if foil crimped together and stack on two other pieces of foil done the same way. I make a 'lid' out of two more stacked pieces. ChucksI cook till tender--I do not temp them but will check at the 4-hour mark. The thicker roasts that I prefer tend to run about 5 hours total. The way I do chucks is described
here.
Butts, as noted, I don't foil till they come off. At that point I'll flip them onto a double thickness of HD foil, wrap tightly and rest.