Prick duck skin all over with the tip of a knife. In a heavy skillet brown kielbasa and duck in batches over moderately high heat, transferring as browned to paper towels to drain. Combine kielbasa and duck with broth and water in a 6 gallon kettle and bring to a simmer.
Pour off all but 1/4 cup fat from skillet. Add flour and cook roux over low heat, stirring constantly, until a shade darker than peanut butter, about 30 minutes. Add onions, celery, and bell peppers, and cook, stirring occasionally, until vegetables are crisp-tender. Add vegetable mixture to kettle and stir until roux is dissolves.
Simmer gumbo, uncovered, for 2 hours and let cool completely. Bone duck, discarding skin and bones. Chill gumbo overnight. Gumbo may be made 2 days in advance and kept covered and chilled. Discard fat on surface and reheat gumbo with cayenne, scallions, and salt over moderate heat. Serve over rice.