In 1781 he was made Baron Conyngham, of Mount Charles in the County of Donegal, with remainder to his nephew Francis Burton, and Earl Conyngham, of Mount Charles in the County of Donegal, which like the creations of 1753 and 1756 was created with normal remainder to the heirs male of his body.

He was the eldest son of Mary, sister of the first Earl Conyngham, by her husband Francis Burton.

Two of his sons, Victor, the fifth Marquess (who was aide-de-camp to Lieutenant General Sir John Grenfell Maxwell during World War I, and who died on 9 November 1918 in York), and Frederick, the sixth Marquess, both succeeded in the titles.

Francis Burton (died 1743), father of the second Baron Conyngham, was a Member of the Irish Parliament for Coleraine and County Clare.

The Honourable Sir Francis Burton, second son of the second Baron, was a colonial administrator and Member of Parliament for County Clare.

Henry Francis Conyngham, Earl of Mount Charles, eldest son of the first Marquess, represented County Donegal in the House of Commons but predeceased his father, unmarried.

Lord Francis Conyngham, second son of the second Marquess, was Member of Parliament for County Clare.

The heir apparent is the present holder's son Alexander Burton Conyngham, Earl of Mount Charles (born 1975).