Combine chicken, stock, beans, tomatoes, onion, bell pepper, corn, chiles, garlic, chili powder, cumin, salt, and black pepper in a 5- to 6-quart slow cooker. Cover and cook on low until chicken is very tender, 5 to 6 hours.
Transfer chicken to a cutting board; shred into small pieces; return to slow cooker. Stir in lime juice.
Divide soup evenly among 5 bowls. Top evenly with yogurt and cilantro.