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

Yes. Because of daylight saving time the gap moves between +7h and +8h.

When is the best time to call Karachi from Auckland?

Between 16:00 and 17:00 in Auckland, which is 09:00 to 10:00 in Karachi.

Auckland is 7h ahead of Karachi. This gap is not fixed: daylight saving time moves it between +7h and +8h during the year.

Auckland

New Zealand

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Karachi

Pakistan

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A 9-to-5 day in each city overlaps for 1h. That is 16:00–17:00 in Auckland, which is 09:00–10:00 in Karachi.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
Karachi · Pakistan	Asia/Karachi	UTC+5

What is the time difference between Auckland and Karachi?: Auckland is 7h ahead of Karachi. This gap is not fixed: daylight saving time moves it between +7h and +8h during the year.
Does the time difference between Auckland and Karachi change during the year?: Yes. Because of daylight saving time the gap moves between +7h and +8h.
When is the best time to call Karachi from Auckland?: Between 16:00 and 17:00 in Auckland, which is 09:00 to 10:00 in Karachi.