Assuming you have it, have you contacted your travel insurance company regarding the costs incurred. You may be covered by your insurance

As far as Emirates goes, it is not accountable for the flights to Dunedin. If you had purchased a ticket from Emirates to take you all the way to Dunedin, then it would then have been responsible for rebooking your flights and for providing accommodation. A ticket combining the flights to AKL and the onward flights could have been purchased from a travel agent.

In this case, once it delivered you to Auckland, its responsibilities ended

With separate tickets , having to clear customs/immigration and getting to the domestic terminal, 2 hours is not enough time to allow for any delays that could occur

Looking at the schedules, EK412 is due to arrive at 13:00 and NZ has flights to Dunedin as late at 17:55 sun-fri and 16:00 on sat. Since the last flight was at 16:00 , I assume that yourarrival into AKL was last saturday - EK412 arrived at 14:45 , so a total of 1hr 45 minutes late. Even EU261 , if can be construed to handle onward delays, would not provide any compensation for such a short delay

Really you should have booked to travel the next day , since 3 hours is just not enough time to allow in AKL on separate tickets

Changing crews at Sydney is normal procedure ; it is a legal requirement for Australia that all passengers disembark at Sydney; it is not permitted to remain onboard during the connection

Although too late to be any use this time, you would have been better off booking flights to Christchurch for getting to Dunedin ( EK does offer service to CHC), since if you did miss the 18:00 flight from Christchurch , at least it is driveble to Dunedin in about 5 hours