Drivers and dynamics of initial human migrations across individual islands and archipelagos are poorly understood, hampering assessments of subsequent modification of island biodiversity. We developed and tested a new statistical-simulation approach for reconstructing the pattern and pace of human migration across islands at high spatiotemporal resolutions. Using Polynesian colonisation of New Zealand as an example, we show that process-explicit models, informed by archaeological records and spatiotemporal reconstructions of past climates and environments, can provide new and important insights into the patterns and mechanisms of arrival and establishment of people on islands. We find that colonisation of New Zealand required there to have been a single founding population of approximately 500 people, arriving between 1233 and 1257 AD, settling multiple areas, and expanding rapidly over both North and South Islands. These verified spatiotemporal reconstructions of colonisation dynamics provide new opportunities to explore more extensively the potential ecological impacts of human colonisation on New Zealand’s native biota and ecosystems.