Note that anywhere I note a presumed date, I am basing it off the apparent pattern of basing present-day settings in the year following the year of publication.
The first Christmas special saw the first full length episode featuring David Tennant. On Christmas Eve, Earth is invaded by the Sycorax, all while The Doctor is still recovering from his regeneration at the end of the first season.
Once again set in London, future regular companion Donna Noble is miraculously teleported into the TARDIS, while she walks down the aisle. The Doctor is left shocked at how someone has managed to teleport into the TARDIS, especially while in flight.
This episode is set in a recreation of the ship, The Titanic, as it visits London for Christmas. The Doctor has to save The Titanic, which is on the verge of plunging into Buckingham Palace.
This episode was part of a series of specials that released between seasons three and four. The Doctor travels back to 1851, and runs into another regeneration of himself; but all is not quite as it appears.
This episode set could be counted as a Christmas special; it was released in two parts, with the first episode released on the 25th of December, and the finale released on the 1st of January. It marks the final episode for David Tennant, and the (brief) return of The Master.
Season Six: A Christmas Carol - Sardicktown, located on an unknown planet, somewhere around the 44th century
This is the first Christmas special to be set far into the future, and the first to feature Matt Smith. The Doctor must convince the "scrooge" of this tale, Karzan Sardick, to allow a ship of people to land safely. Unfortunately, Karzan does not want a bar of it, and would rather let them die.
After the Doctor is helped out of a tricky situation, he returns to repay the favour, by helping the newly widowed Madge Arwell keep her children's spirits up over Christmas. This does not go according to plan, when they are teleported to a strange planet.
The Doctor retires to Victorian England, having lost Amy and Rory Pond, and having decided to travel alone. Things do not go to plan, when strange snowmen start to appear, and he runs into a familiar face.
The Doctor teams up with two of his previous incarnations to prevent another alien invasion. David Tennant returns as an incarnation of The Doctor, and John Hurt appears as "The War Doctor"; the incarnation of The Doctor who burned Gallifrey.
Special: The Time of the Doctor - The UK, some time around the 21st century (presumably 2014), and Trenzalore, during an unknown time.
The Doctor is drawn to a planet centered around Christmas, when all of his enemies are drawn to the one place, heeding a mysterious broadcast. This episode marked the last episode for Matt Smith.
The Doctor, Clara and Santa must race against time to save a base of scientists, who have become the victims of brain-eating dream-inducing alien crabs. This is the first Christmas episode to star Peter Capaldi.
The Doctor reunites with River Song, who mistakes him for a real doctor, having not met this regeneration before. This episode marks the assumed last appearance of River Song, ending in the fated date referenced when we first meet River Song, in Silence in the Library.
When a little boy accidentally swallows alien technology, granting him super powers, the Doctor returns to enlist his help in saving the world from yet another potential alien invasion.
Series Eleven: Twice Upon a Time - South Pole, 1986, Ypres, 1914, Villengard, unknown time (possibly 1986).
When the Doctor refuses to regenerate, he finds himself running into his first ever incarnation, who is also refusing to accept his next regeneration. This episode is notable for several reasons, including the last episode to star Peter Capaldi, the inclusion of Peter Bradley as the first doctor, the brief return of Jenna Coleman as Clara Oswald, the inclusion of The Captain (ancestor of long-running companion, The Brigadier), and as the first episode to star Jodie Whittaker as the Thirteenth Doctor.