I don't know of any comparison that has explicitly compared the two but here are two angles you could look into:

• Nicholas Dirk's Castes of Mind argues that the modern caste system (the four varnas, etc., not the fact that people were born into hundreds of tiny jats) is largely a product of British colonialism. You could compare attempts to rationalize race in America, first the extensive classifications of "mulattos" ("octaroon", etc), and then the elimination of a white - mixed - black triracial caste to one that was white - one drop biracial system. Other areas still have a triracial (well, quatraracial with indigenous people or immigrants being the fourth race) system, for instance South Africa and Brazil. (Edward Bonilla-Silva thinks we're headed back for a triracial system, but that's something else). The move from triracial to biracial system was particularly stark in Louisiana where "free people of color" (gens de couleur libres), many of who were Créole/"Créoles of color", had long flourished. I'm not well versed in this literature, but judging from title, maybe White by Definition: Social Classification in Creole Louisiana is a place to start. Further, you can explore the literature on how various ethnic groups in America (Irish, Italians, Jews) became "White" over the course of the 19th century. This a more narrow topic, and would require a lot of reading about America, but it's one way to go.

I don't know precisely what your assignment is but one of my favorite books is about the status of Muslims in India.

• Muslims in Indian Cities edited by Laurent Gayer and Christophe Jaffrelot is one of the most interesting books I've read in the past five years. In American sociology, there's long been a difference between a ghetto on the one hand and a slum. In a ghetto, you traditionally had lots of different classes, but only one ethnicity. In a slum, you can have a lot of ethnicities, but only one class (the poor). Traditionally, in areas like the South Side of Chicago, the Black middle class had to live along side the Black underclass (Loic Wacquant and others call the current situation, where the Black middle class has moved out of former ghettoes, the "hyperghetto"). These authors use this concept that was used to look at the plight of Black people in America to look at the plight of Muslims in India. It's a very interesting and well researched book because they manage to get ethnographers to look at most major cities with Muslim populations. Each chapter is by a different author. The two write especially about Amedabad in Gujarat, likely the city where Muslims have it worst and the most ghettoized, but other writers about very different cities. They are explicitly doing a comparison between the American and Indian situations, but they are comparing the American racial situation to religious ("communal") rather than caste situation in India.