For sauce, in a 1-1/2-quart saucepan, combine ketchup, corn syrup, white vinegar, brown sugar, onion, mustard, Worcestershire sauce, garlic, black pepper, hot pepper sauce, cumin or chili powder, and cayenne pepper. Bring to boiling; reduce heat. Simmer, uncovered, for 25 to 30 minutes or until thickened, stirring occasionally.
Cut the ribs into serving-size pieces. Sprinkle Rib Rub evenly over all sides of ribs; rub in with your fingers.
Drain wood chips. In a grill with a cover, arrange medium-hot coals around a drip pan. Test for medium heat above pan. Put some of the drained mesquite chips onto coals. Place ribs, bone sides down, on grill rack over pan. Cover and grill for 1-1/2 to 1-3/4 hours or until ribs are tender, adding more coals and remaining mesquite chips as needed. Brush ribs with some of the sauce during the last 10 minutes of grilling. Pass remaining sauce. Makes 4 to 6 servings.
In a blender container or small food processor bowl combine dried rosemary, dried thyme, dried minced onion, dried minced garlic, coarse salt, and ground black pepper. Blend or process until coarsely ground.