In a small bowl, whisk the oil, vinegar, lemon juice, mustard and garlic until combined; season with salt and pepper.
In a shallow dish or large bowl, combine the mozzarella, tomato, and basil. Pour marinade over and toss well to coat; refrigerate for at least 30 minutes and up to 2 hours. Add bread cubes and let stand 5 minutes.
Grease grill grates and heat to medium. Thread skewers alternating between mozzarella, tomato wedges, cherry tomatoes, basil leaves, and bread cubes; reserve marinade. Grill the skewers 7-10 minutes, turning a few times, until the cheese is soft and everything is heated through. Serve alone or over a bed of lettuce with reserved marinade on the side