Francis succeeded his uncle, Frederick Drake-Brockman, in the Beechborough estate, as the latter had no male children. His grandparents were James Drake-Brockman, High Sheriff of Kent, and his wife Catherine (Tatton). His great-grandfather was The Rev. William Tatton, D.D., Rector of Rotherfield, Sussex, and Prebendary of Canterbury Cathedral. Francis' son, William, succeeded him in the Beechborough estate.