What happens when a scheduled job interview gets rescheduled?

You go in at the rescheduled time and proceed with courtesy and professionalism. If you are the one who had to reschedule, apologize for the inconvenience and thank them for the consideration of your emergency (surely it was an emergency that caused the change!). If they were responsible for rescheduling, and you know the reason why, perhaps mention it once (I certainly hope you are feeling much better now, I was sorry to hear of your being ill) but, DON'T criticize them for you're change of plans.