How were missionaries able to convert native populations? What made it possible for Christian missionnaires to convert natives of America, Africa and Asia to another religion despite the fact that they already had beliefs and gods ?
The power of the Holy Ghost

In "Preach My Gospel: A Guide to Missionary Service" (the manual used by LDS missionaries) it states (in Chapter 4)


  The power of the Holy Ghost. The witness that comes to sincere
  investigators before baptism comes through the power of the Holy
  Ghost. “The power [of the Holy Ghost] can come upon one before
  baptism, and is the convincing witness that the gospel is true. It
  gives one a testimony of Jesus Christ and of his work and the work of
  his servants upon the earth” (Bible Dictionary, “Holy Ghost,” 704).
  The Holy Ghost testifies of truth. All people can know the
  truthfulness of the Book of Mormon through the power of the Holy
  Ghost. “By the power of the Holy Ghost ye may know the truth of all
  things” (Moroni 10:5).


This applies to both native and non-native populations that missionaries have taught or are currently teaching.

Of course, this is not the only method used by Christian missionaries throughout history. Some Churches in the past have relied on coercion, unfortunately. But sincere and lasting conversion can only come via the power of the Holy Ghost.