Ashmore was born in Lincolnshire on 6 September 1907. When he was a teenager, his family relocated to Smethwick in the West Midlands, where his father had work in the horse trade. Fred and his brother Joe began racing together in 1935, initially sharing an MG, forming the partnership that would define their early careers in motorsport.
In 1936, Fred and Joe Ashmore entered into a business arrangement with Reg Parnell — who would become one of Britain's most prominent racing drivers and team managers of the postwar era — to found Highfield Garages in Derbyshire. The commercial ties between the Ashmores and Parnell would have a direct bearing on Fred's racing career over the following years. Separately, Ashmore operated his own garage business, Ashmore Auto Engineers, in West Bromwich, though by this period he was living on a farm near Lichfield.
Ashmore took part in the major British racing events from 1947 to 1949. His initial mounts were various ERA (English Racing Automobiles) models that Parnell and the Ashmores had accumulated during the Second World War years, when racing had been suspended and pre-war machinery could be acquired relatively cheaply.
From 1948 onward, Ashmore moved predominantly to Maserati 4CLTs, cars that Parnell entered for several drivers. The entries were sometimes listed under the name Scuderia Ambrosiana, an Italian entrant with which Parnell had a mutual arrangement covering the logistics of shipping cars and sharing prize money across heavily regulated postwar borders.
Ashmore was named as a reserve entrant for the 1948 British Grand Prix, reflecting his standing within the Parnell-adjacent racing fraternity. That same year he produced a solid result at the Penya Rhin Grand Prix in Spain, finishing sixth overall in a Maserati 4CLT. The result raised his profile sufficiently that he was given entries at three national-level Grands Prix in 1949: the Swiss, British, and Belgian rounds. At Silverstone for the British Grand Prix and at Spa-Francorchamps for the Belgian Grand Prix, Ashmore finished sixth, consistent performances in machinery that, by 1949, was competitive in British and continental national-level events. The Maserati 4CLT he drove at those events was one previously used by Alberto Ascari, who would go on to win the Formula One World Championship in 1952 and 1953.
Ashmore did not compete in any FIA Formula One World Championship events, which began in 1950. His final Formula One appearance was at the 1951 Richmond Trophy, where he drove Parnell's ERA B-type — an elderly car dating from the pre-war period and carrying the provenance of having been raced by the Thai nobleman B. Bira. Engine problems prevented him from starting the race, and this proved to be the end of his racing career.
After retiring from competition, Ashmore returned full-time to his farm near Lichfield. In 1971 he sold a portion of the farm land for the construction of a motorcycle racing circuit. He was married for more than 50 years; his children became champion show jumpers, maintaining the family's connection to competitive equestrian sport that dated back to his father's involvement in the horse trade.
Ashmore's brother Joe also raced. Joe's son Gerry Ashmore competed in Grands Prix during the 1960s, continuing the family's presence in motor racing into the next generation. Another of Joe's sons, Chris, raced at lower formulae level.
Fred Ashmore died on 22 November 1989 at the age of 82.