The title passed down through generations, with various earls serving in the House of Lords as Scottish Representative Peers and holding other political positions.

The 8th Earl received the Victoria Cross and held political office as Captain of the Honourable Corps of Gentlemen-at-Arms.

The barony of Dunmore became extinct after the 9th Earl's death in 1980, but the other titles passed to his distant relatives in Tasmania, Australia.

The family seat was located at Amhuinnsuidhe Castle on the Isle of Harris and Dunmore Tower near Falkirk.

He was made Lord Murray of Blair, Moulin and Tillimet (or Tullimet) and Viscount of Fincastle at the same time, also in the Peerage of Scotland.

Murray pleaded guilty but received a pardon from King George II and succeeded to the peerages when his brother died unmarried six years later.

The fourth Earl was a Scottish Representative Peer in the House of Lords from 1761 to 1774 and from 1776 to 1790 and served as colonial governor of New York, Virginia and the Bahamas.

In 1839, the people of South Harris were ejected from their homes by armed soldiers and a posse of Glasgow policemen acting on orders from the government, at the behest of the Earl of Dunmore.