Grant, the son of a genteel Buckinghamshire family, was transported to New South Wales on the Coromandel, leaving England on 5 December 1803 and arriving in Port Jackson in May 1804. He had originally been sentenced to death for the attempted murder of a solicitor who had interfered in his courtship of an aristocrat's daughter, but was reprieved and his sentence was commuted to transportation for life. Accorded special status and liberty because of his gentlemanly background, Grant is considered to be the first among the colonial population to have written serious poetry, his earliest known effort being The Chicken and the Grasshopper.
Grant was active in the life of the colony. After befriending the wealthy malcontent (and convict) Sir Henry Browne Hayes, he was radicalised, and became increasingly outraged by what he perceived as government perpetrated injustices, a theme that recurs in his surviving letters. Eventually, he was tried for sedition and sentenced to five years hard labour on Norfolk Island. Grant was later transferred to Van Diemen's Land before being returned to Port Jackson in 1808. During his final years in the colony, Grant spent much time with James and Anna Maria Lewin assisting them in the publication of their work on the birds of New South Wales.
Grant was finally pardoned by Governor Macquarie and sailed for England in 1811 on the Spring Grove.