I am extremely hesitant to attempt an answer to this question at all because of the extreme complexity, ambiguity, and modern relevance inherent in it. Different people, including myself, will give radically different answers to this question depending on their own inherent biases and cultural background. Compounding the problem is the sheer number of different cases that were handled and treated by various societies in the Americas. If we ignore those of mixed Indian-African heritage, we may broadly categorize people of mixed heritage into Indian-American (often considered either White or Indian depending on social factors), French-Indian (Métis), and Spanish-Indian (Castas).

Each of these groups were treated specially, particularly in regions where they were common. However, I want to stress that each of these groups has been the subject of numerous books and intense study. In the particular case of the castas, I must note that there remains a vibrant and active discussion on the integration of Mestizos and other mixed-heritage Mexicans into the United States.

In spite of how complex this debate is, we do have a wonderful fixed date for the beginning of this integration: The ratification of the Treaty of Guadalupe-Hidalgo. The end of the Mexican-American war necessitated the integration of the former Mexican citizens into wider US society.

This process was marred by conflict from the start. From even before the end of the war, there had been vigorous debate as to the potential status of Mexicans in the conquered territories. I give some background on this debate and sources in this post. In the end, the Mexican system was too distinct from the American system to be integrated. One of the most important distinctions to note in the Mexican political landscape was the breakdown of the Spanish caste system with Mexican independence. This breakdown was especially evident in the far-flung northern reaches later annexed by the US and was consolidated in the Plan of Iguala, which guaranteed legal equality for all Mexicans and established the independence of Mexico. The US had no such institutionalized ideals and in practice the newly naturalized Mexican-Americans were disenfranchised within wider US society. Rather than responding to new racial orders in the annexed territories the US imposed the racial systems of their older territories. These were characterized by a delineation between "white" and "indian", with little distinction for the broad range in between. This hierarchy was necessarily extended to incorporate the largest population of Mestizos as a single group that was not quite "white", but was above "indian". Mestizos who fell into this group quickly learned to emphasize their European heritage. The former Mexican Indians who could not plausibly claim near European heritage quickly lost all claims to their land in short order. In California's constitution, the right to vote was restricted to the "White Male citizen" and "White Male citizen of Mexico". When the Arizona Organic act was passed in 1863, the California voting requirements were incorporated into its territorial constitution.

While all of that is an [extremely] brief overview of mixed-race enfranchisement for one group, the broad strokes are similar for other groups. The concept of "whiteness" was integral to 19th century racial hierarchies. Those groups that did not fit these conceptions (including slavic, irish, jewish, indians, and sometimes even italians) were subject to disenfranchisement. But many or most of the ethnic groups in the US were mixed race. For one extreme example, most of the Eastern Native American tribes possessed a majority of members with at least some European heritage. It's important to note that US racial hierarchies were not entirely based on this lineage, but rather on ambiguous racial classifications that often did not line up with heritage.

^1 "Chicano Indianism: A Historical Account of Racial Repression in the United States" Martha Menchaca *American Ethnologist * Vol. 20, No. 3 (Aug., 1993) , pp. 583-603

For a more in-depth introduction to the latin american racial systems, I vaguely recommend Race and ethnicity in Latin America. That subject is enormously complex so any introduction is necessarily flawed. Nevertheless, this is a decent place to start.