Prepare the slow-cooker. If using agave leaf, roast over an open flame or under a broiler (being careful not to touch any of the exposed flesh of the leaf), turning frequently, until charred and pliable, 10 to 15 minutes. Lay in the bottom of the slow-cooker. If using avocado leaves, scatter half of them on the bottom of the slow-cooker and set the others aside. If using banana leaf, drape it from side to side in the slow-cooker. (If it feels too stiff to drape without cracking, you can soften it by slowly pulling it across the flame or coil of a burner.)
Prepare the meat. Heat enough oil to coat the bottom of a very large (12-inch) skillet over medium-high. (If your slow-cooker has a stovetop-safe insert, you can use it to brown the meat—without the leaves, of course.). Pat the meat dry with paper towels, salt it liberally and lay it in the pan. Brown on all sides—this will take 10 to 15 minutes, depending on the size and shape of the meat and pan. (Beef cheeks will have to be done in batches.) Transfer to the slow-cooker (or, if using the cooker’s insert, remove the meat and place in the leaves).
Make the marinade and cook the meat. Combine all the marinade ingredients except the beer in the blender with 2 cups water and blend until smooth—this will take a minute or two because of the chiles’ tough skins. Pass through a medium-mesh strainer into the slow-cooker, using enough to coat all sides of the meat. (For 3 pounds of meat, you’ll have a scant cup left to refrigerate and use as a grilling marinade.) Pour in the beer, if using. Cover with the remaining avocado leaves or fold the banana leaves over the top, if either of those was your choice. Cover and turn on the slow-cooker to high. Your barbacoa will be done in about 4 hours, though you can hold it longer. (My slow-cooker can be programmed to switch from high after 4 hours to a “keep warm” temperature for up to another 6 hours. You may have to switch yours to “keep warm” manually.)
Serve. Coarsely shred the meat and arrange it on a warm platter. Skim any fat from the sauce that remains in the slow-cooker, then taste it. If you feel the brothy sauce would be better with a more concentrated flavor, pour it into a medium saucepan (or, for stove top-safe inserts, simply remove the insert), set over high heat. Boil for a few minutes to reduce the quantity, then taste and season with salt if it needs it. Spoon as much as you like over the lamb or beef (I like to sprinkle a little coarse salt over the meat at this point.). Serve the barbacoa with the tortillas, salsa and/or hot sauce, onion, cilantro and lime.
No slow-cooker? In a large (6- to 8-quart, at least 12 inches in diameter) heavy pot (preferably a Dutch oven), combine the ingredients as described. Set the cover in place and braise in a 300-degree oven for about 3 hours, adding water occasionally if necessary to ensure the liquid stays at about the same level. Finish as outlined above.