Cover urad dal with at least 3 inches of cold water in a bowl and refrigerate for 8 hours; this soaking time is included in prep time. Drain and rinse.
Place soaked dal, canned kidney beans, tomatoes, onion, garlic, ginger, cumin, garam masala, turmeric, cayenne, salt and 3 cups fresh water in a 6-quart slow cooker.
Cover and cook on low for about 8 hours until the whole dal is very soft. Mash about 1 cup against the side of the cooker and stir it back through the pot.
During the final hour, bring rice and 1 1/2 cups water to a boil in a covered saucepan, reduce to low and cook for 40 to 45 minutes, then rest covered for 10 minutes.
Stir butter into the dal and turn off the cooker. Let bubbling stop for 5 minutes, then whisk yogurt with 1/2 cup hot dal and stir the tempered mixture back in.
Divide rice and dal among 6 bowls, adding a spoonful of the thick cooking liquid to each portion.