Use a general database (recommend Scopus or Web of Science) to find health literature on your country and immigrant group.
Example: Scopus search for Cuban immigrants and health
Limit your search to "Review" articles to find secondary literature. Review articles will provide good background info. You can always toggle back to include "Articles" to view the primary literature.
You might also limit the dates to the most recent research.
You may find it helpful to sort by "Relevance". You can toggle between options (Date, Cited By, Relevance).
To find additional research on the sociological aspects of your immigrant group's experience try searching SocINDEX, Race Abstracts, and Global Health.