You will also need to have a New Zealand version of the Airpoints card which has a Flybuys barcode on the back of the card to be able to scan at Flybuys retailers. You can't present an Airpoints card without the barcode because it wouldn't work.
As soon as you change your address to New Zealand, they will send you a replacement card with the barcode but obviously you will need to have a New Zealand postal address and do it in advance.