Hello there. I can mostly answer the part of your question about African-Americans and the CPUSA, but I can provide some additional information as well. The part of your question I can not really answer is the last question, about Mexican and Chinese immigrants. I apologize that I cannot get to that.

The shortest answer is that interest in leftist organizations among communities of color increased throughout the first half of the 20th century. However, support varied depending on which organizations we are talking about exactly, and also which minority group specifically. We can spend all day discussing the many different leftist parties and their particular views on race, but I am going to focus on two of the biggest and most influential in early 20th century America: the Communist Party USA (CPUSA, founded 1919, Marxist), as you mentioned, and the Socialist Party of America (SPA, founded 1901, democratic socialist).

Historically speaking, communist movements around the world have a dedication to being anti-racist; at least ideologically anti-racist, to account for outliers. If they are a Marxist organization this is especially so, as Marx himself wrote a good deal about the issue of race and came to the conclusion that the unity of being working-class is far more important than any racial difference. The CPUSA attracted many members of minority groups due to its stance against racism. Non-Marxist socialist organizations did not have this guarantee (this is more of a fun fact: Nguyễn Tất Thành, the man who became Hồ Chí Minh, looked to the communist movement in the early 20th century for this very reason). In fact, while the SPA had some history of anti-racist delegation, it was a very hot topic in the organization in the early 20th century, many members were segregationists, and the SPA did not make serious or successful effort to expel these elements from the Party. Eugene Debs, one of the most influential and successful SPA members ever, was a staunch anti-racist and frequently encouraged the SPA to adopt better anti-racism policies.

Since the SPA had inconsistent support of leftist people of color, lets look at the CPUSA and the African-American community, as your post asked about. Generally speaking, the CPUSA had a much better success rate with recruiting people of color than the SPA did during this period, especially African-Americans; this was largely due to their different stances on the issue of racism as mentioned before. From its founding in 1919 until 1928, the CPUSA stayed ideologically committed to anti-racism, however they did not put much effort into large-scale recruiting efforts of people of color. In 1928, the CPUSA began to attempt to recruit African-Americans by adopting the idea that the Black community of the United States was a distinct national group due to the particular conditions of their oppression, and went as far to say that the Black community should gain its own state in the United States. However, this change was more ideological and didn't change their practices much, as such it did not boost their popularity in African-American communities much either. This position was held until 1935, when the CPUSA changed their policy and began to promote a "Popular Front" against fascism, solidarity among leftist organizations to fight against fascism and all it represented, and this included racism. At this point, the CPUSA began serious direct action to promote anti-racism, such as unionizing black workers around the country, and had improved relations with the NAACP. This is when the relations between the CPUSA and the African-American community were at their height, at least in this part of the century, about 1935 to 1939. African-American support for the Party was affected by the Molotov-Ribbentrop Pact when it was signed in 1939, and it is known that African-American membership did drop due to the CPUSA's support of it; they were accused of being soft on Nazis, essentially. The CPUSA held onto the position that their support of the pact was an anti-war stance, not wanting any of the nations to go to war at all. After World War II, the CPUSA struggled to remain particularly relevant, and by the 1960s and 70s Black communists would much sooner join the Black Panther Party, which was also Marxist, rather than CPUSA.

I hope you found this answer useful! I'm sorry I could not give you much information in the way of the Mexican and Chinese immigrants, I do not know the history of that as well.

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Edit: Qualified the sentence "Generally speaking, the CPUSA had a much better success rate with recruiting people of color than the SPA did, especially African-Americans; this was largely due to their different stances on the issue of racism as mentioned before." by adding "during this period."