The applicable Chauffeur Drive terms & conditions are those of the carrier operating the flight for which the Chauffeur Drive service is used to get to or from the airport. This applies regardless of whether your ticket may be a codeshare ticket issued by a different carrier.
Itineraries which are exclusively between New Zealand and Australia, on Emirates-operated flights, are not eligible for Chauffeur-drive. All other itineraries on Emirates-operated flights are eligible, including Business Class itineraries which involve a stopover in Australia.