The body mass index (BMI) is a rough estimate of fat mass (although it can be confounded by muscle mass), and is the weight in kilograms divided by the square of the height in meters [BMI = weight / (height^2)]. A BMI of 25 to 30 is considered overweight; 30 and over is considered obese.
The graphs I'm about to present require some explanation. The data in each graph were collected from the same individuals over time (15-69 years old). That means some weight gain is expected, as this population normally gains weight into middle age (then loses weight). What's interesting to note is the difference in the rate of weight change between migrants and non-migrants. The first two data points in 1968 are baseline, and compare non-migrants with "pre-migrants" still living on Tokelau. The second two data points in 1981-82 compare the same individual migrants in New Zealand with the same non-migrants.
Unless they all decided to become body builders, migrants to New Zealand gained more fat mass than Tokelauans between 1968 and 1982. The rate of weight gain in New Zealand was more than twice as fast for men and more than 50% faster for women than on Tokelau.
Why did Tokelauans and especially migrants to New Zealand gain weight? Probably because they had greater access to a wide variety of calorie-dense, palatable foods of modern commerce. The introduction of wheat and sugar, at the expense of coconut and traditional carbohydrate sources, was the main change to the Tokelauan diet during this time period. See this post for a graph.
Finally, there's the question of exercise. Did a change in energy expenditure contribute to weight gain? The study didn't collect data on exercise during the time period in question, so all we have are anecdotes. During this time, men living on Tokelau progressively adopted outboard motors for their fishing boats, replacing the traditional sails and oars. Their energy expenditure probably decreased.
But what about women? Tokelauan women traditionally perform household tasks such as weaving mats and preparing food. Their energy expenditure probably didn't change much over the same time period. Since both men and women on Tokelau gained weight, it would be hard to argue that exercise was a dominant factor.
How about migrants to New Zealand? Here's a quote from Migration and Health in a Small Society: the Case of Tokelau:
Overall it is our belief that most of the migrants expend greater energy in their work than is currently the case in Tokelau.Exercise doesn't appear to have been the main factor, although the data don't allow us to be totally confident about this.