His father was William Moffat and his mother was Helen Dobson. The third of eight children, he was born in Glasgow, Scotland. There, he was exposed at an early age to the theatres and sixpenny galleries of the city and to numerous famous comedians and entertainers of the day.
He was married in 1897 to the stage actress and costume designer, Margaret Liddell Linck (1882–1942). His wife was also a suffragist, and in 1907 Graham Moffat founded the Men’s League for Women’s Suffrage in Glasgow. He also penned a suffrage propaganda play, The Maid and the Magistrate.
Bunty Pulls the Strings was a huge hit in London’s West End, his biggest success, running for 617 shows at the Haymarket Theatre in 1911. In reviewing the play when it debuted on Broadway late in 1911, the New York Times classed Moffat beside J. M. Barrie and wrote, "it is the freshest and most wholesome thing that the theatres in New York have housed since the days of Peter Pan." The Scottish actress Molly Pearson played the title role of Bunty in New York. The comic actor James Finlayson played Bunty's father both New York and in London. The play was made into a film in 1921.
Moffat himself directed at least one film, Till the Bells Ring, a 46-minute experimental sound film based on the comedy of the same name he mounted in 1908. The film starred Moffat, his wife and his daughter.
Besides his plays, he published at least three books:
- The Pawky Scot (1928), a book of Scottish humour, with illustrations by Arthur Moreland.
- Towards Eternal Day: the Psychic Memoirs of a Playwright (1950) in which he revealed his belief that his dead father and brother helped from the dead in writing his plays. He also wrote on the supposed return from the dead of Sarah Bernhardt.
- Join Me in Remembering (posthumous, 1955) incomplete upon his death and finished by his daughter.
- Scottish Theatre Archives at Glasgow University
- Clan Moffat Genealogy
- Margaret Moffat at Complete Index to World Film
- Elizabeth Crawford, The Women's Suffrage Movement: a reference guide 1866-1928, Routledge, 1999
- New York Times, "Evening Of Rare Joy With Scotch Players In Graham Moffat's Delightful Comedy, When Bunty Pulls the Strings October 11, 1911
- Bunty Pulls the Strings (1921) at the Internet Movie Database
- Till The Bells Ring (1933) at British Film Institute. This source says the film was made in 1926 and shown in a private showing of Perfectly Synchronised Talking Motion Pictures on 28 January 1926 at the Holborn Empire.