Dawnay was the son of John Dawnay, 1st Viscount Downe, by his second wife Dorothy, daughter of William Johnson, of Wickham, Lancashire.
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He was succeeded in the viscountcy by his grandson, Henry, his son the Honourable John Dawnay having predeceased him.
John Dawnay, 4th Viscount Downe (1728–1780), MP for Cirencester and Malton, son of the above
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John Dawnay, 5th Viscount Downe (1764–1832), MP for Petersfield and Wootton Basset, son of the above
Dawnay died in October 1695, aged 70, and was succeeded in the viscountcy by his son Henry.