Slice sausage links into 1/2-inch chunks. Cook sausages in a skillet over high heat until browned but not cooked through, 3 to 5 minutes. Remove and pat dry with paper towels to remove excess fat. Transfer to a slow cooker and add chicken broth, sweet potatoes, jerk marinade, garlic, thyme, and cinnamon.

Cook on High for 3 hours. Add black beans, reduce heat to Low, and cook for an additional 2 hours.