Does the time difference between Auckland and Dubai change during the year?

Yes. Because of daylight saving time the gap moves between +8h and +9h.

When is the best time to call Dubai from Auckland?

There is no overlap between standard working days, so one side has to call early or late.

Auckland is 8h ahead of Dubai. This gap is not fixed: daylight saving time moves it between +8h and +9h during the year.

Auckland

New Zealand

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Dubai

United Arab Emirates

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A 9-to-5 working day in Auckland does not overlap one in Dubai at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Dubai · United Arab Emirates	Asia/Dubai	UTC+4

What is the time difference between Auckland and Dubai?: Auckland is 8h ahead of Dubai. This gap is not fixed: daylight saving time moves it between +8h and +9h during the year.
Does the time difference between Auckland and Dubai change during the year?: Yes. Because of daylight saving time the gap moves between +8h and +9h.
When is the best time to call Dubai from Auckland?: There is no overlap between standard working days, so one side has to call early or late.