Known as F. R. Kempson, he was born at Stoke Lacy in Herefordshire to William Brooke Kempson (1796 - 1859), the rector of Stoke Lacy, and his wife Elizabeth (née Roberston).

[2][3][4] His Herefordshire secular building designs include Hereford Museum and Library, built in 1873–74 in Venetian Gothic style;[5][6] Sir James Rankin's house, Bryngwyn Manor, at Much Dewchurch in 1868; the restoration of Burton Court in Eardisland in 1865; and Cheyney Court at Bishop's Frome in 1870.

[7][8][9][10] His works in Wales include St Peter's Church in Pentre, with fellow architect John Bacon Fowler.

[11] Kempson married Julia Madeleine Jay in 1866; the marriage produced six children.

[1] His granddaughter Rachel (1910–2003), was married to the actor Sir Michael Redgrave.