There are a plethora of oral histories available on the Internet. Do some searches for the country of origin, ethnic groups or specific experiences of the people you are researching for more!
The Library of Congress has digitized a huge collection of primary resources. Use the menu on the left to narrow down your choices, or search go back to the search box to look for specific materials/peoples/keywords.
This is one of the oldest and largest civil rights organizations advocating for Latinx people in the U.S. Click 'Control F' and search for 'migration' or other related terms to search the page.
George I. Sanchez was an educator, civil rights activist and writer who dealt with issues of education for the children of migrants, especially in the U.S. Mexico border region.
This is the official archive of the Voces Oral History Project. You can find overview and some interviews at https://voces.lib.utexas.edu/, but the archive contains the full text of all interviews.
The Benson is the archive for NPR's StoryCorps Historias project, documenting Latinx stories from around the US, many of which involve immigration/migration stories from Mexico, Central America, South America and the Caribbean.