The tune usually used is the American Civil War song "The Bonny Blue Flag." Siggy Spaeth tried to link that tune to an Irish tune called "The Irish Jaunting Car." There several tunes & songs that are currently known as "The Irish Jaunting Car," but none of them sounds remotely like "The Bonny Blue Flag." The song writer was Harry McCarthy. That's name historically associated with the border area of counties Cork & Kerry. It might be fruitful to dig into old tunes from that area.