Looking for Parish info on John Mulligan

Looking for information for John Mulligan who left Donegal for Liverpool and eventually landed in Quebec City, Canada.  He eventually settled in Onslow, Quebec.  We toured Southern Ireland in 2011 not realizing that our ancestors were from the North.  We want to tour Northern Ireland this summer or next.  We purchased a document on the "Mulligan" while in Ireland which states that Mulligan (O MAOLAGAM) originated in the Baronies of Boylagh and Raphoe but that the name is also found in Ulster, Leinster, Connacht.  Counties Dublin, Monaghan, Mayo and Sligo being the most favoured locations.  Therefore we are unsure as to what parish to join to find the most information on our ancestors.  Any and all information would be appreciated.

Thank-you.