Heat oil in a large saucepan over medium heat. Add onion and cook, stirring, until softened, about 2 minutes. Stir in garlic, cumin and chile powder and cook for 30 seconds. Add tomatoes and lime juice; bring to a boil. Reduce heat to a simmer and cook until the onions are very soft, 16 to 20 minutes. Stir in turkey (or chicken) and beans and continue cooking until the mixture is heated through, 3 to 5 minutes more. Divide the turkey-bean mixture among tortillas (or wraps). Top each with cheese and cabbage, roll into burritos and serve.