On the day of the shoot, when it's Sir Richard's turn to be with Mary, they are arguing and waiting for Richard's gun man to show up. They are standing in the spot where Richard it to shoot from, but it's an impossible spot because it's in a grove of large trees. Shooting must be done out in the open where the birds can fly by. They will never fly where the canopy above is the limbs of giant trees. Matthew is in his spot out in the open and he has to walk into the grove of trees to get to them.
In the scene where Thomas meets Lord Grantham after his dog has been returned, Thomas's collar is shown undone and hanging loose. It then disappears entirely, then reappears, then disappears and then appears tucked away on the other side of his coat.
When Anna takes Rosamund and Lady Mary upstairs to walk in on Lord Hepworth and Rosamund's maid, Shore, Anna is seen first opening the door, and then entering the room with Rosamund and Mary. She is present throughout the scene.
However, after the confrontation with Hepworth, when Rosamund and Mary turn to leave, Anna is still seen behind them, about to leave the room as well and close the door behind them.
But when Mary and Rosamund are seen in the hallway a moment later, Anna, who was just seen about to close the door and walk away from them down the corridor has completely disappeared and is nowhere to be seen.
The door can also be heard being pulled shut as Mary and Rosamund are seen standing in the hall, however Anna, who closed the door is seemingly nowhere in sight.
At 1:31, Anna and Daisy begin to push the pointer across the Ouija board. Anna just has two fingers on it. When the camera zooms in on close-ups of Anna's and Daisy's hands, she has four fingers on the pointer. At 1:32, the camera backs away and just her two fingers are shown on it again.
Sir Anthony says he has lost the use of his right arm. He holds his left out to shake hands but the shot of his arm returning to his side shows his right arm. (The shot of his arm returning to his side is from a different camera angle. The right arm in blue belongs to Edith.)
Thomas mentions doing the Black Bottom to the dowager countess as they are about to dance. It's 1920. The Black Bottom was not popular until the mid 1920's, nobody in 1920 would have understood the reference. It's a fast dance, not at all suited for an older woman like Violet. Thomas was trying to be funny.
One the New Years Day shoot, every time a bird flies over Matthew fires a single shot, the ejects and replaces TWO cartridges.
Though stated as being a planchette that the staff was going to try, they were in fact using an Ouija board, the successor to the planchette.
The miniature lights that adorn the Christmas tree by the hundreds at the beginning of the episode -- which is supposed to take place during Christmas of 1919 -- were not introduced until 1970.
Modern Christmas lights visible in Christmas tree (beginning of episode).