Why did Marcus Garvey's return to Africa movement lose steam, as well other ventures? And gave rise to intergrationalist/equal rights as preached by Dr. Martin Luther King
The "Back to Africa" ideology, modernly thought of as originating in large part with Marcus Garvey, is a much older idea than most think. This idea to return to their roots is largely cyclical, losing steam and then gaining prominence every few generations as the situation worsens or betters for Black Americans. But this is an ideology that has been an undercurrent to larger African-American thought since the beginning. First in the late 1700s (possibly earlier, as more evidence is being discovered all the time,) then kickstarting again in the 1820s with the American Colonization Society, dying out for a bit in the 1830s-40s and rising again in the 1850s, dying out during the Civil War and once again rising during Reconstruction.
The ideology really picks up again in the 1920's with the UNIA's Back to Africa movement under Marcus Garvey and his Black Star Line. Personally, i'm not of the opinion that Garvey's BtA Movement held sway for any great period of time, due to Garvey's deportation (due to mail fraud charges) and Liberia's refusal to work with the UNIA (Liberian dealings with the US government), but it is important to note. Garvey is just one more piece in a long line of prominent activists pushing to find a separate homeland, be it Africa, Haiti or Canada.
In regards to the second part of your question, i'm not entirely sure that the death of Garvey's movement gave rise to integrationist thought, the death of Jim Crow and Segregated society had long been a goal of African American activist thought, with the idea of leaving somewhat more of a subset (I like to think of it as a Plan B) to the larger goal of being American, in America, as opposed to American in Africa. Much of this thought comes from the common sense argument that "I'm born here, raised here, know the people and culture. Why the hell would I want to move to Africa/Haiti/Canada/Wherever?" Its much easier to stay and fight for your rights than uproot everything and leave to go somewhere you've never been and may never even make it to.
I can elaborate pretty decently on other Emigration plans (With primary sources too!) but if you have any additional questions. I'll be more than happy to answer.
Sources
Negro With a Hat: The Rise and Fall of Marcus Garvey by Colin Grant is one of the better Garvey biographies out there that goes into the UNIA and the ideology behind it to a much deeper extent.
The Marcus Garvey and Universal Negro Improvement Association Papers Project
Marcus Garvey's Back to Africa movement wasn't well accepted into the white supremacist communities of America. Its not explicitly stated as to why, but in The Autobiography of Malcom X, the reaction from the Klu Klux Klan and the Black Legion('A radical offshoot of the KKK')to Malcom's father's support of the movement and the Universal Negro Improvement Association, led there to me violence against his family, ending with his father's death. No reason is given to lead to an opposition of the movement from white supremacist groups(such as the KKK and Black Legion), except for quotes from the Autobiography of Malcom X:
The Klansmen warned her that we had better get out of town because ‘the good Christian white people’ were not going to stand for her husband ‘spreading trouble’ among the ‘good’ Negroes of Omaha with the ‘back to Africa’ preachings of Marcus Garvey.”
(More reasons can be found in TheWireMother's comment)
Similarities between the UNIA(Universal Negro Aprovement Association) and NAACP(National Association for the Advancement of Colored People) can reflect the ideologies of the integration:
The Objectives of the UNIA:
To establish a universal confraternity among the race.
To promote the spirit of race pride and love.
To reclaim the fallen race.
To promote a conscientious Christian worship among the native tribes of Africa.
To establish universities, colleges and secondary schools for the further education and culture of the boys and girls of the race.
To conduct a worldwide commercial and industrial intercourse.
Objectives of the NAACP:
The NAACP's principal objective is to ensure the political, educational, social and economic equality of minority group citizens of United States and eliminate race prejudice. The NAACP seeks to remove all barriers of racial discrimination through the democratic processes. In essence, the ideologies of the UNIA changed from a more separatist view, to a more integrational view that was adopted by the NAACP.
Sources:
The Autobiography of Malcom X(as told by Alex Haley)
In addition to OP's question; was there any interaction between civil rights leaders and the government or people of Liberia during this time?
Garvey was sent to jail on a charge of mail fraud, and the resulting power vacuum left the Universal Negro Improvement Association in a shambles, and doomed the Black Star Line (at that point already crippled by embezzlement, infighting, and political sabotage).
Garvey was a significant early target of the "anti-radical" section of the Bureau of Investigation (which later became the FBI), headed up by J. Edgar Hoover, and was just one of the many political leaders of the time who were deported, incarcerated, or killed as a result of politically-motivated FBI investigations. The dissolution of Garvey's political base and the Palmer Raids a few years before foreshadowed the FBI's relationship to the civil rights movement 35-50 years later (via CoIntelPro).
Garveyism as a movement never really recovered after Garvey's incarceration. A LOT happened in the black community between Garvey's incarceration and Dr. King's ascension - The Great Migration, Brown v. Board of Education (and other cases fought and won by the NAACP), the rise of the Nation of Islam, etc; relative to a lot of the other gradual changes in America, King's rise was swift. A big part of the sudden nature of his rise was his capitalizing on the success of the Montgomery Bus Boycott, and the fact that it allowed him to unite more than 60 southern black ministers to form what would become the Southern Christian Leadership Conference, subsequently, almost overnight uniting thousands of parishioners with shared interests under one banner. It helped their image in the eyes of the American white establishment that a large segment of SCLC advisory and membership was white, and it was far easier to sell the relatively peaceful idea of "equality" over the potentially threatening idea of "empowerment," which had been the message of a lot of pro-black political movements up to that point. The role of religion was also significant, in that King appealed to a largely Christian America using their own scripture to support his argument - differing greatly, for instance, from the markedly foreign-seeming NOI, who it was easy to demonize based on religion alone.
Well, the final nail must have been when the American colonists descendents in government of Liberia were executed on the beach in 1980.