I've added yogurt to soap batter, in an ungelled cold process soap, for a simple teaching recipe of olive, soft oils, coconut & sugar.
Gently gelled milk soaps are very nice.
Powders are very easy to work with (add to the soap batter directly, or dissolve in water and add).
Whey works as a water replacement and doesn't heat too much.
Milk heats if it's a water replacement (whether it's fresh, or reconstituted powder), but is nice.
Whether you add lactate by powder, whey, yoghurt or milk itself, take care with temperature control (it is a very different soap, scorched).
All of the milk fats I've tried have added a smooth creaminess to the finish feel of the soap.