There's an Irish "rocking the cradle/baby that's none of my own" song which was played on uilleann pipes by Peadar Broe and I think also by Leo Rowsome. It's supposed to refer to St. Joseph. Not sure if I see any obvious connection to the tune that I remember for "Git along..." though.

The tune to "Streets of Laredo" is know in Ireland as "Bold Phelim Brady/The Bard of Armagh", though I seem to remember seeing a comment on a thread some months ago indicating that there were older English (British) words to the same tune.

When I was a child, a commercial recording of SoL appeared in Ireland, and I can still remember my parents' outrage at "those damned Americans robbing an Irish tune". They wouldn't have had much knowledge of or respect for folksongs, which to them were just "Comeallya's"!