Ashmore came from a racing family, his garage-owning father Joe and uncle Fred having raced private Maseratis in partnership with Reg Parnell immediately after the war. Along with his brother Chris he raced Formula Junior in 1960. Later that summer he traveled to Europe and raced at Zeltweg and Innsbruck, finishing third behind Hans Herrmann and Wolfgang von Trips. In 1961 he raced a Reg Parnell Lotus 18 and later with The Three Musketeers team, finishing second to Giancarlo Baghetti in the Naples GP at Posillipo. He took part in the British GP that summer with a privately-entered Lotus but retired after just a few laps. Later in the year he traveled to Europe again and raced at the German and Italian GPs but without success, his Monza outing ending with a big accident. He reappeared at Monza in a Lotus in 1962.