Very interesting observation and thanks for this question.
There may be quite a few possibilities. One could be that Yes, there may have been a migration of Indic (from the Indian subcontinent) people thousands of years ago (at least according to Vedic mythologies and puranas). Who these people were, how they reached the other end of the world, etc; there has been no proof such as ships or ship routes/land routes found. Even if there has been any evidence, it may be under the oceans now as deluges over thousands of years may have sunken the land mass (look at the land mass between Australia and India, which has proven the movement of some Indian and Australian tribes between these two countries).
Ancient flight paths? Seems a little far stretched. However, it may not be, if we believe in the 'Ancient Alien Theory'. Again, it may sound ridiculous but not so much after piling evidence over years.
The second possibility is that there may have been a central civilization (call it Atlantis or whatever) which must have been the root of all these major civilizations (almost proven according to Archaeology. Look at Gobekli Tepe, Turkey) After the last major ice age about 12,000 years ago, when the central civilization got destroyed, the remnants may have moved to the neighboring land masses and must have started all over while still retaining the culture. So, the 'Vedic' culture as well as the Andes culture could have common roots.
A third and a very practical possibility is that there has been significant movement of Indians to the Americas in modern times. In fact, almost 43% of Guyana's population is Indian. And South America indeed has a measurable Indian population in many countries. So, it could be quite possible that you may have heard Vedic hymns.