The history of Mormon race-relations is a complicated one. In the early years of the Church's history, when the Church was still led by Joseph Smith, the overall attitude by Church members towards race could be characterized as fairly liberal for the time. Most early Mormons were from northern states and a fair number of them were abolitionists. This even exacerbated the conflicts between Mormons and non-Mormons in the state of Missouri, with pro-slavery southern settlers in Missouri fearing the political influence of such a large influx of mostly anti-slavery northern Mormons.

Joseph Smith himself seems to have been fairly liberal for the time in his perspective on race. He ordained black men to the priesthood, and his platform for his presidential campaign called for the government to purchase and free all the black slaves currently owned in the south.

Race relations in the LDS Church began to change with the death of Joseph Smith and the change in leadership to Brigham Young. Brigham Young lacked the liberal race attitudes of his predecessor and largely set the tone for Mormon perspectives on blacks for the next century. He declared the priesthood inaccessible to black members and permitted slave owners immigrating from slave states to retain their slaves in Utah territory. The rationale behind this was that blacks were "cursed" due to being the descendants of either Cain or Ham, whose unrighteous actions in the Old Testament caused them and their posterity to forfeit the right to hold the priesthood and placed a mark of blackness on their skin.

This perspective became the dominant one for the next century. Over time, as the national perspective on race began to shift, Church members began to rationalize why God would have a seemingly racist policy in place and would punish people for their ancestry or the color of their skin. Many Mormons embraced the idea that blacks were somehow "less valiant in the preexistence." To explain, this centers around the Mormon belief in a "preexistence" or an existence as spirits prior to coming to Earth. In this preexistence, spirits could choose to follow God's plan to come down to Earth or could rebel and follow Satan. Many Mormons, including prominent scholars like Bruce R. McConkie theorized that blacks were denied the priesthood because they had been "fence-sitters" in this conflict between God and Satan. This helped to rationalize the seemingly arbitrary and racist exclusion of a sizable chunk of humanity based solely on their ancestry/skin color.

To answer your last question: one of the things that complicated LDS missionary work with "non-white" people was that the racial teaching was very specifically targeted toward African ancestry.

This challenged missionary work in Brazil, for example, where LDS proselyting was very successful, and it was sometimes unclear whether a person with dark skin was of African or Native American ancestry. While Africans were prevented from holding the priesthood, Native Americans were actively sought out, due to the general belief that Native Americans were of Israelite descent. According to LDS historian Alonzo Gaskill, this kind of arbitrary hair-splitting was part of what spurred the Church to abandon its racial policy.

As far as how this was explained to "non-white" races, that gets even more interesting. From the 1950s through the 1970s, there were small congregations in Africa that petitioned the leadership of the Church to send missionaries. These people weren't rebuffed -- they were told to "wait".

Some of the apostles (the highest leadership of the Church, below the prophet) believed in the teaching that people of African descent were "less valiant" in the pre-existence and would never receive the priesthood, but others adopted theories that there was some kind of temporary purpose for the ban (as when Jesus forbade his disciples to preach to the Gentiles in Matthew 10:5-6). From that point of view, the black Mormons who waited for the ban to be lifted were compared with Cornelius, the Roman centurion who moved Peter to open the preaching to the Gentiles (Acts 10); or the Canaanite woman who asked Jesus for a blessing even after he called her a "dog" (Matthew 15).

The negative racial sentiments at least did not extend to the peoples of the various islands in Oceania, some of them have the highest percentage of Mormons by total population. Mormon churches were established there well over 100 years ago. Tonga has the highest percentage of Mormons, at about 57%. According to the church, their Tonga church was established there in 1891.

Regarding Polynesian areas. They were taught the same doctrines as was taught throughout all North and South America. That the tribe of Ham (African Americans) were not eligible for the Priesthood. Polynesians were not considered the tribe of Ham. They were also considered by the church to be descendants of Hagoth (Machaa) and Corianton (Hawaii Loa)