Mr Graham, 39, and his family will move from Perthshire and he takes up his position at the BSJA in Stoneleigh in August.
BSJA chief executive Di Cornish said: “His professionalism, depth of knowledge and experience across equestrianism as a whole will prove extremely valuable as we continue to move forward.”
Mr Graham has worked for BE for six years and was tipped for the job of BE sport director when Mike Etherington-Smith was promoted to chief executive in November 2008.
But instead BE appointed a commercial director and two sport managers (news, 12 February).
Mr Etherington-Smith said: “It is sad that we are losing such an experienced and professional regional director but it is a great opportunity for him.”
Mr Graham also chaired the BE safety committee for two years and represented BE in Scotland as director of facilities and events for the Scottish Equestrian Association.