This is not an easy question to answer, because statistics are not great.
Starting in the 1840s, there was a influx of immigrants. Most of them came from England, Australia and other parts of Europe like Ireland and Germany. Part of the appeal of New Zealand is that it is a natural resource rich country. During this time some Chinese came as well. They came in search for wealth during the Otago Gold Rush, which began 1861.
The Chinese population did not grow, because the New Zealand government levied a poll tax on the Chinese up until the 1930s. This, as one can imagine, hindered immigration. In light of this, to the best of our knowledge, all we can say is that the first real Asian immigration took place in the 1860s.