The son of Sir Robert Cunliffe, 2nd Bt., and his wife, Mary Wright, Sir Foster succeeded his father as Baronet in 1778. On 1 October 1781, he married Harriet [d. 1830], daughter of Sir David Kinbloch, Bt. They had four surviving sons, of whom the second son, Sir Robert Henry Cunliffe, succeeded his father as 4th Bt.