Ashby was born on 17 April 1744 at Wotton-under-Edge, Gloucestershire, and was apprenticed to a clockmaker in that town, who also engraved dial-plates, spoons, and tankards.
On the termination of his apprenticeship he removed to London, where, following the bent of his inclination for writing-engraving, he entered into an engagement with Mr. Jefferies, geographer, of Charing Cross, his principal employment being to engrave titles for maps and charts.
Subsequently, Ashby worked for John Spilsbury, writing-engraver, of Russell Court, Drury Lane, to whose business he eventually succeeded, and whose widow he married.
Ashby was employed by provincial, colonial, and foreign bankers, to engrave notes and bills.
Among the works for which he engraved the plates are: In his later years, Ashby lived in retirement at Exning, Suffolk, where he died on 31 August 1818, at the age of 74.