Preheat oven to 350º. In a large, heavy pot or Dutch oven over medium heat, toast guajillo, morita, and pasilla chiles, stirring frequently, until softened and fragrant, 2 to 3 minutes.

Transfer chiles to a medium heatproof bowl, then cover with boiling water. Using a small plate or bowl, completely submerge chiles. Let soak until rehydrated and pliable, about 20 minutes.

Meanwhile, generously season brisket and oxtails with salt and pepper. In the same pot over medium-high heat, heat oil. Working in batches, cook beef, turning occasionally, until browned on all sides, 6 to 7 minutes per side for brisket/roast, 4 to 5 minutes for bone-in parts. Transfer to a plate or cutting board.

In a blender, blend chiles, 1 1/2 cups chile liquid, tomatoes, garlic, cloves, cinnamon, vinegar, cumin, and oregano, adding more chile liquid if needed, until a pourable paste forms, about 1 minute.

Return beef to the pot and add onion, bay leaves, and chile paste. Pour in enough water to just cover beef; season with a heavy pinch of salt. Bring to a simmer over medium heat.

Remove from heat, cover, and bake until beef is fork-tender, 4 to 4 1/2 hours. Discard bay leaves and onion, transfer beef to a cutting board, and reserve broth (or consomé). Shred beef with 2 forks.

Taste consomé; season with salt and pepper. Depending on your preference, to thin out consomé, add water or chicken or beef stock until the desired consistency is reached.

In the same large pot over medium-low heat, bring consomé to a bare simmer (there should be a layer of dark red-looking fat/oil on top).

Construct a taco station: Have the consomé simmering on one burner, a large cast-iron skillet or plancha heated over medium heat on another, a plate of shredded beef from the birria, and another plate of tortillas to the side, as well as small bowls filled with Oaxaca cheese, onions, and cilantro.

Working in batches, reheat 1/3 cup beef on one side of cast-iron skillet, leaving enough room for a tortilla, tossing occasionally, until seared and browned. Take a tortilla and dip it into consomé, coating both sides with dark red fat. Place tortilla on cast-iron skillet next to beef and cover with cheese.

Fry the tortilla until cheese is mostly melted and the underside has browned and started to crisp, about 3 minutes. Place seared meat on one half of tortilla and top with onion and cilantro. Fold tortilla into a taco and cook until crisp, about 30 seconds per side. Repeat with remaining beef.

Serve tacos with lime wedges and small bowls of consomé for dipping alongside.