An advert in an magazine led him to apply for a job as an assistant for "an international cartoonist"; this was on 'Flash Gordon' and Asbury moved to Austria for six months before clashes with Dan Barry led to his departure. Back in England he designed cards for Hallmark, rising to become their chief designer. Married in 1969, he decided to go freelance and found work drawing for D. C. Thomson's Bunty. With the launch of Wizard in 1970 he graduated onto boys' adventure strips, drawing 'Soldiers of the Jet Age', 'The Crimson Claw', 'The Secret of Deep 16' and others for the paper. At the same time, he also found work on Joe 90: Top Secret, soon to merged with TV21, where he drew 'Forward from the Back Streets' and 'Tarzan'. Some brief-run strips in Countdown led to him drawing 'Cannon' for TV Action and TV Comic before he was hired by Look-In, where, after briefly drawing 'Follyfoot', he had his first big hit with 'Kung Fu'.
At the same time, Asbury took over the artwork for 'Garth' in the Daily Mirror following the death of Frank Bellamy. He was to draw the strip for 21 years, working initially with Jim Edgar; later scriptwriters included Angus Allan, John Allard, Tim Quinn, Phil Harbottle and, from 1995, Asbury himself.
In the early days of the strip, Asbury was also able to continue working for Look-In, his strips for that paper including 'Dick Turpin', 'Battlestar Galactica' and 'Buck Rogers in the 25th Century'. However, an opportunity arose in the early 1980s for a change in artistic direction.
"For Greystoke I did nearly 3,500 huge drawings, many of them in full colour. I didn't know they were going to be fed through a copying machine and come out as grey blotches. I learned my lesson on that.
Since the release of Greystoke in 1984, Asbury has storyboarded dozens of movies, a few sample credits would include Labyrinth, Willow, Alien 3, Chaplin, Interview with the Vampire, Fierce Creatures, Quills, Lara Croft: Tomb Raider, Thunderpants, The Hours, Troy, Alexander, Batman Begins, The Da Vinci Code, The Boat That Rocked, the last six James Bond movies (Brosnan/Craig) and Harry Potter and the Chamber of Secrets.
Although Asbury's work on 'Garth' was his most widely syndicated, he is probably most fondly remembered for Look-In and the dynamic look he brought to 'Kung Fu' and 'The Six Million Dollar Man'. He continues to work as a storyboard artist, his most recent work being for the upcoming Between Two Worlds.