To make the sauce: In a medium saucepan, over medium heat, warm the oil. Add the garlic, chile powder, and cumin. Cook for 30 seconds, stirring occasionally. Add the remaining sauce ingredients, whisk them together, and allow the sauce to simmer for about 5 minutes.
Remove the thin membrane from the back of each rack of ribs. Allow the ribs to stand at room temperature for 20 to 30 minutes before grilling.
To make the rub: In a small bowl, mix the rub ingredients. Season the ribs all over with the rub, pressing the spices into the meat.
Drain the mesquite chips and toss them onto the burning coals or into the smoking box of a gas grill. Grill the ribs over Indirect Low heat (grill temperature should be about 300°F) until the meat is very tender and has shrunk back from the ends of the bones, 1 1/2 to 2 hours. About 15 minutes before the ribs are done, start brushing occasionally with the sauce on both sides.
Transfer the ribs to a baking sheet and tightly cover with aluminum foil. Let rest for 30 minutes before serving. Serve warm.