Son of Alexander Murray (d. 1750) of Broughton and Euphemia Stewart, daughter of James Stewart (d. 1773), 5th Earl of Galloway, and Catherine Montgomerie (d. 1757), daughter of the 9th Earl of Eglinton. Married Catherine Stewart in 1752. Catherine Stewart was a sister of Keith Stewart and John Stewart, 7th Earl of Galloway. Murray's mother was a sister of Catherine Stewart's father, making Murray and his wife first cousins.
Murray was MP for Wigtownshire (1762-1768) and Kirkcudbright Stewartry (1768-1774). Ca. 1785 he "scandalized his friends by absconding abroad with [his niece, Grace Johnston], refused to desert her, and abandoned all further political ambitions."1
James Murray built the impressive Cally House for himself in 1763. It is today a 56 room hotel known as Cally Palace.2
Murray and his wife arrived at Patrick Heron's Kirroughtrie estate on September 22, 1762. Boswell was staying there at the time while on his Harvest Jaunt of that year. On this occasion, Boswell described Murray as "a most amiable man, [who] has very good sense, great knowledge of the world, and easy politeness of manners." On September 23, he and Murray had a long talk about Boswell's idea of seeking a commission in the guards. Murray recommended Lord Adam Gordon for introduction into life and Boswell did go visit Gordon shortly after arriving in London in that same year.