He was born at Bristol on 8 July 1743, into a Catholic recusant family: he was the fourth son of Sir John Swinburne, 3rd Baronet of Capheaton, Northumberland, who married on 20 July 1721 Mary, only daughter of Edward Bedingfeld, and granddaughter of Sir Henry Bedingfeld of Oxburgh, Norfolk.

He afterwards studied at Paris, Bordeaux, and in the Royal Academy at Turin, devoting special attention to literature and art.

[1] Swinburne met in Paris his future wife Martha, daughter of John Baker (1712–1779) of Chichester, solicitor to the Leeward Islands, who was being educated at a convent of Ursuline nuns.

[1] Swinburne in the company of Sir Thomas Gascoigne travelled through Spain, returning to Bayonne in June 1776.

[1] On his return to Bayonne in June 1776 Swinburne, with his family, travelled to Marseilles, and a supplementary volume describing the expedition was issued in 1787.

At Vienna Maria Theresa conferred on Mrs. Swinburne the female order of La Croix Étoilée, and the Emperor Joseph II stood godfather to their son of that name.

With letters of introduction to the French court, from Vienna, Swinburne went to Paris (1783), and through Marie-Antoinette' s influence obtained a grant of all uncultivated crown lands in the island of St. Vincent valued at £30,000.

In February 1785 William Pitt offered half that sum for it, and on receiving a refusal passed through parliament a bill to impose heavy taxation upon the unproductive lands in all the West Indian islands.

[1] In the meantime Swinburne was sent to Paris in September 1796 as commissioner, to negotiate an exchange of prisoners with France, but, in the face of difficulties arising from the capture by the French of Sir Sidney Smith, was unsuccessful, and in December 1797 was recalled to England, In December 1801 he went out to the lucrative post of vendue-master in the newly ceded settlement of Trinidad, and also as commissioner to deliver up the Danish West Indian islands to a Danish official.

[1] There were published in 1841, under the editorship of Charles White, two volumes entitled The Courts of Europe at the close of the last Century, which consisted of the letters of Henry Swinburne, mostly on foreign life (dating from March 1774, and chiefly addressed to his brother, Sir Edward Swinburne).

[5] Another daughter, (Marie Rosa) Louisa, befriended the widowed Princess Elizabeth of the United Kingdom, who had married in 1818 the future Frederick VI, Landgrave of Hesse-Homburg.

The second son, also Henry, born 1772, was enrolled among the French royal pages, and placed under the care of the Prince de Lambesc.