McKay and Duursma make up half of the Dons' trade and free agency additions from a busy off-season signing window, with the pair moving to Tullamarine from North Melbourne and Port Adelaide respectively.
The Bombers' entire squad returned to pre-season training this week with a number of new faces starting their preparations for the 2024 campaign.
Recruits McKay (knee) and Duursma (hip) won't take part in main training for the coming weeks as they deal with minor issues, with the new Essendon duo set to work through tailored training programs ahead of the new year.
The Bombers are likely to have Dylan Shiel (knee), Sam Draper (groin) and Sam Weideman (Achilles) back into main training next year as they recover from their respective ailments, with Shiel's return date likely set for February.
Young forward Tex Wanganeen isn't expected to be at full fitness until February after the 20-year-old required an operation on a navicular injury following the 2023 season.
Talls Nik Cox (ankle) and Harrison Jones (back and ankle) are expected to gradually be integrated into main training as early as this week.
Essendon's 2024 season is set to commence on Saturday, March 16 against Hawthorn at the MCG in Round 1.