Here is further evidence for some of the points I raised in my article on punishment. If I had to guess, one reason this child has issues with bullying is that she has learned to be a bully from the example of her father. The “lesson” he is teaching her is that it is ok to bully! While making her walk to school could be considered a “natural consequence” of her getting kicked off the bus, and so not a punishment at all, it very much depends on the mind set of her parents. I worry that the underlying cause of her bullying could be a history of harsh punishment.
My commentary on punishment: Here, and above.