[1] Following his father's sudden premature death soon after August 1919, his mother took the family back to Edinburgh where Dempster then entered George Heriot's School and where he was active in rugby and cricket.

[1] After qualifying and a brief time working as a locum general practitioner, Dempster joined the Royal Air Force and served in India and Burma during the Second World War.

[3] At her suggestion, he applied and was accepted to Ian Aird's surgical unit at the Postgraduate Hospital, Hammersmith, as a researcher in organ transplantation, investigating the outcome of dog kidney allografts.

[1][3] He worked at the Royal College of Surgeon's Buckston Browne Farm, with Sir Arthur Keith, an anatomist and anthropologist, and jokingly referred to the job as one of the hospital's worst roles.

Simultaneous work by Morten Simonsen [de] and René Küss found that placing the donor kidney in the pelvis was preferable to a superficial site and all concluded that an immunological mechanism was responsible for rejection.

[4][5] Subsequently, Dempster became acquainted with international organ transplantation peers including Georges Mathé of Paris, who also believed that immunological reactions explained graft rejection.