His paternal grandfather, also named Francis Burton (1640–1714), sat in the Irish parliament for Ennis from 1691 to 1714.
[2] Originally a branch of the Musards, Lords of Stavely, the Burton family settled in Richmond, Yorkshire after the Norman conquest.
His grandfather was Major General Henry Conyngham, who claimed significant lands in County Meath, including Slane Castle.
[citation needed] When Earl Conyngham died childless on 3 April 1781, all the titles expired, except the barony, which Burton inherited.
[2][6] The second baron died in 1787 and was succeeded by his eldest son, Henry, a politician and courtier who was elevated to Viscount Conyngham of Mount Charles in 1789.