I also have Falvey family connections. I'm quite sure my family came from somewhere in Co Kerry. Jeremiah and Michael Falvey came first to MA then to RI. They married Sullivan girls, sisters of my great great grandmother, Mary Margaret Sullivan. There was an Ahern family also who were very close to the family and served as sponsors for baptisms and witnesses for marriages.
Michael Falvey, born 1834. His family: His father Hugh 1800 and wife Mary Sweeney, their son Michael Falvey married Catherine Sullivan, children: Mary, Catherine, Hugh, Hannah, Ellen and John Joseph.
Jeremiah Falvey, born 1826 married Joanna Sullivan: his father was Jeremiah, mother Julia. Jeremiah and Joanna's children: Daniel, John, Ellen, Julia, Jeremiah, and Mary. If you ever come across a Hugh and Mary Sweeney and a Jeremiah and Julia circa 1800 in Ireland I sure would appreciate any info.
Sullivans in Kerry are sure lots of fun to research. I have two sets of Kerry Sullivans to search! Like looking for "Smith" in America. Good luck with your Falvey lines.