Stir cherries, broth, preserves, ketchup, vinegar, chipotle peppers, thyme and allspice in a small deep bowl. Transfer to a shallow non-reactive dish or sealable bag, large enough to hold chicken. Add the chicken and turn to coat well. Cover and marinate in the refrigerator for at least 2 hours, overnight is better.
Preheat grill to high. Oil the grill by placing folded paper towel in tongs and rub it over the racks. Remover chicken from the marinade. Transfer the narinade to a medium skillet.
Reduce the grill heat to medium and grill the chicken until cooked through and no longer pink in the middle, 7 to 9 minutes per side. Meanwhile, bring the marinade to a boil. Reduce heat to a simmer and cook until the sauce is reduced by about half, 12 to 15 minutes. Let the chicken cool slightly; serve with the sauce.