He was succeeded in the viscountcy by his grandson, Henry, his son the Honourable John Dawnay having predeceased him.
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Dawnay was the son of John Dawnay, 1st Viscount Downe, by his second wife Dorothy, daughter of William Johnson, of Wickham, Lancashire.
Dawnay died in October 1695, aged 70, and was succeeded in the viscountcy by his son Henry.