My Cunninghams were Roman Catholic, and based on the PATRICK TOHALL about Ulster Migrants to Mayo, it lists both a John Cunningham and a Edmond Cunningham who left Drumgore to Ballina area in 1795 due to sectarian violence.
The Tithe in Kilcummin parish, county Mayo lists a Pat and Edmond paying tithe in Cashel townland, Kilcummin Parish in 1832.
By the time of Griffiths, there are John Sr, John Jr, Peter, Thomas, James all living in Cashel & Castlemagee townlands (they are neighboring). From there, I have the paper trail of Church records, imigration documents and death certificates.
Connecting from Griffiths back to the Tithes back to Armagh is more circumstantial evidence, though ...