This war was named after Lord Dunmore who was appointed Governor of Virginia in the region where the war took place.
The Virginia militia attacked the Shawnee in Kentucky, taking Fort Pitt on the western edge. The war was fought ostensibly to ensure white settlers to spread into traditional Shawnee hunting grounds.
The real reason for the war, however, may have been to divert attention of the Virginians away from disagreements they had with royal appointees to govern them such as Lord Dunmore. Fort this reason the war - 1774, is deemed by some to be the actual foray into the Revolutionary War.
Stevenson, Mark A. (ed) - Merriam-Webster's Collegiate Encyclopedia. Springfield: Merriam-Webster: 2000, p. 967