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.
As a title of example, in the Americas, most Jesuits missionaries shared in the daily life of the Indians, travelling with them as they moved from one encampment to the next.
The two religious traditions took different approaches to evangelism. Catholics, such as the tireless Father Pierre-Jean De Smet SJ (missionary in the old Oregon Country), sought simple expressions of faith from Indians. Protestant missionaries, on the other hand, sought not only to convert Indians to Christianity but also to convert them to a new lifestyle centered on individual labor and community-building. - Missions and missionaries
The Jesuits tried their best to understand the Native languages that they were dealing with.
St. John de Brebeuf noted that in order to be effective as a missionary, one had to master the Native language at hand.
To explain the low number of converts, Brébeuf noted that missionaries first had to master the Huron language. His commitment to this work demonstrates he understood that mutual intelligibility was vital for communicating complex and abstract religious ideas. He believed learning native languages was imperative for the Jesuit missions but noted that it was so difficult a task, that it consumed most of the priest’s time. Brébeuf felt his primary goal in his early years in New France was to learn the language.
He translated Ledesma's catechism from French into Huron, and arranged to have it printed. It was the first printed text in that language (with French orthography). He also compiled a dictionary of Huron words, emphasizing translation of religious phrases, such as from prayers and the Bible.
Brébeuf is credited with composing the "Huron Carol", Canada's oldest Christmas song, written around 1642. Brébeuf wrote the lyrics in the native language of the Huron/Wendat people. The song's melody is based on a traditional French folk song, "Une Jeune Pucelle" (A Young Maid). - St. Jean de Brébeuf (Wikipedia)
One last little note, I would like to add here is that the Jesuits incorporated to some degree native languages into their liturgies:
Algonquian and Iroquoian Uses
Also called "Indian Masses", a number of variations on the Roman Rite developed in the Indian missions of Canada and the United States. These originated in the 17th century, and some remained in use until the Second Vatican Council. The priest's parts remained in Latin, while the ordinaries sung by the schola were translated into the vernacular (e.g., Mohawk, Algonquin, Micmac, and Huron). They also generally featured a reduced cycle of native-language propers and hymns. At present they are rarely used. Latin liturgical rites (Wikipedia)
St. John de Brebeuf, like the Good Shepherd, died a martyr in order to set an example for the others.
When the assault began, the two Jesuits raced among the men, shouting encouragement, urging them to pray, tending the wounded, baptizing the dying. The Iroquois prevailed, capturing Brebeuf, Lalemant and sixty Huron warriors, whom they beat and herded to St. Ignace. Brebeuf called on the Huron warriors to stay true to Christ until death.
Stripped naked, the Jesuits were tied to stakes and tortured with fire. Brebeuf suffered from noon until four p.m. on this day, March 16, 1649. When he would not scream out, but continued to pray for the salvation of his tormentors, the Iroquois heated cauldrons of boiling water and poured it over him in mockery of baptism. They cut and hacked his body, and gouged out his eyes. Before he died, they scalped him and cut out his heart, which they ate. Lalemant died of similar tortures early the next morning. - The Iroquois found John Brebeuf Brave