His grandfather, who had succeeded in 1791 to the baronetcy and family estates in Shropshire, previously held by the English branch of the Acton family,.
In 1851 he was made admiral of the fleet, and in 1852, a year before his death, inherited the family baronetcy from his elder brother, being himself succeeded by his brother William, dean of York, who died in 1858.
The Burgoynes of Sutton, whose baronetcy dates from 1641, have been in Bedfordshire since the 15th century, whilst the Osborn family have owned Chicksands Priory since its purchase by Peter Osborn in 1576.
Henry Dymoke was created a baronet; he was succeeded by his brother John, rector of Scrivelsby (1804-1873), whose son Henry Lionel died without issue in 1875, when the baronetcy became extinct, the estate passing to a collateral branch of the family.
The baronetcy to which he had succeeded passed to the son of his brother John, provost of Edinburgh.