You missed an era rather than an episode...
There is NO linguistic and cultural evidence supporting settlement of Polynesia from South America. And there is certainly no genetic evidence.
Heyerdahl's ideas were crackpot views from the 70s that don't survive examination in the light of the evidence. He may have paddled about in the seas in his raft, but we now know conclusively (from linguistic, cultural, genetic, domesticated animal, and pottery evidence) that the settlement of Polynesia originated from Asia (and modern day Taiwan in particular).
There is evidence of CONTACT between Polynesia and South America: the sweet potato (known as the kumara in new Zealand) was a staple food crop in Polynesia but originated in South America. But given the proven sea-faring abilities of the Polynesians at the time, the most likely explanation is that they travelled to South America and brought the kumara back themselves. Remember, these are the same guys who also settled as far West as Madagascar, so reaching South America in the East is not that unlikely.
Cheers