I remember going to a lecture at Otago University several moons ago where the pre-European colonisation of New Zealand was discussed seminar-style by some experts, the names of whom meant nothing to me, this not really being my field.
The theory in vogue at that time, and this may have changed in the 10-15 years since, was that there were two major waves of Polynesian migration to New Zealand, one about 1000-1200 AD and another one later, perhaps c1600. The first wave of migration may have been quite accidental and sparse - canoes being blown off course from time to time. Certainly, at that time there was a fair amount of support for the idea that the Chatham Islands (200 miles of the coast of the South Island) were settled during the first wave. The common word for these people was Maoriori or Moriori. At the time there was little archaeological evidence of their lifestyle, although I understand that they've been busy on this one. Of course, no one knows what their language was, although it was being assumed at the time that it was a variant of whatever form of "Polynesian" was being spoken.
The second wave of migration - the "canoe period" (and Max may be able to give you better info about that) brought a much larger and better organised wave of immigration. There was some debate over whether it was deliberate policy by the rulers of ?the Cook Islands? to relieve overpopulation and minimise warfare. Whatever, it was believed that the canoes (five, Max?) all came from the same place. The language they brought with them became what we now know as Maori - sorry Max, no double letters and no macron. The lack of real linguistic drift apparently supports the thesis that the immigration was quite recent.
Back to the seminar, and here's where there was a lot of controversy which I think still rages in anthropological circles. The newcomers ran headlong into the established land ownership/territorial boundaries instituted by the original migrants. The languages may have been dissimilar enough so that negotiation would have been difficult or maybe nobody really bothered. Although early European historians would have it that the Maori killed and ate the Moriori, it was believed that it was much more likely that there was a gradual assimilation, except in the Chathams. There, it appears that the original inhabitants were left pretty much to their own devices until the mid-18th/early 19th century when a bunch of Tainui invaded. I don't know the details of this.
If the Moriori were not pure Polynesian, a lot would be explained by assimilation in terms of the physiological differences between the Maori and other Polynesians - not great, but noticeable. My opinion here!
I remember one linguistically-inclined lady holding forth at some length in support of the recency of the main migration. She maintained that the linguistic drift in Maori from the original Cook Island Polynesian language would have been much greater if more than a few hundred years had elapsed since the migration.
Since I don't speak either language, I can only report, so FWIW.
