Why are so many islanders obese?

Via Chris Bodenner, Joshua Keating reports:

What really sets the size of these islanders apart is the size of their islands: Tuvalu, Palau, Nauru, and the other countries on the obesity list are among the world's smallest countries in terms of land area and population. So a single tourist resort, fast-food chain, or trade deal has a much more profound effect on society than it would, say, in India or Nigeria.  

Could it be something about Polynesians?  After all, there are some hefty Maori in New Zealand and that is not a small island, especially not measured in economic terms.  Mexico (not Polynesian, of course) also has a growing obesity problem and that cannot be attributed to the island factor.  The same is true of the Persian Gulf states and there Keating suggests very rapid modernization as a culprit.

Keating discusses other factors.  Don't Polynesians naturally eat a starchy diet?  Are island groups more used to the prospect of famine, and thus their bodies store fat more readily?  Being heavy is not low status on many of these islands.  Here are some separate (speculative) claims about their voyaging history.  Nauru is the heaviest island  population, so maybe it has something to do with not having to work for a living, in this case due to phosphates.  Cape Verde and Okinawa are islands, but their residents do not seem to be very heavy.

Don't residents of (some) small islands have weaker prospects of migrating to large cities and might that affect their dietary decisions?  I think of rural isolation as a factor behind obesity, though Keating does not mention that.  Being heavy is also one way of identifying with the local rather than the global culture, and islanders may be faced with stronger pressures to reaffirm their identities.  I would like to see a comparison between Samoans who move to New Zealand and those who stay put.