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

Yes. Because of daylight saving time the gap moves between +9h and +10h.

When is the best time to call Moscow from Auckland?

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

Auckland is 9h ahead of Moscow. This gap is not fixed: daylight saving time moves it between +9h and +10h during the year.

Auckland

New Zealand

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Moscow

Russia

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

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Moscow · Russia	Europe/Moscow	UTC+3

What is the time difference between Auckland and Moscow?: Auckland is 9h ahead of Moscow. This gap is not fixed: daylight saving time moves it between +9h and +10h during the year.
Does the time difference between Auckland and Moscow change during the year?: Yes. Because of daylight saving time the gap moves between +9h and +10h.
When is the best time to call Moscow from Auckland?: There is no overlap between standard working days, so one side has to call early or late.