Crosbie was commissioned as an ensign in the 38th Regiment of Foot in 1757. He was promoted to lieutenant in 1759 and captain in 1769. After serving at the evacuation of Boston in March 1776 during the American Revolutionary War, he was promoted to major in 1778. Promoted to lieutenant colonel in 1781, he became commanding officer of the 22nd Regiment of Foot on promotion. He raised the 89th Regiment of Foot in December 1793 and was promoted to major-general in 1794.