A New Hope - S4-E1
Continuity mistake: After Pete dies and Artie prepares to use the astrolabe to turn back time, Pete's body shifts position on the ground when the camera cuts away but nobody moved him.
All the Time in the World - S4-E19
Continuity mistake: In the Istanbul's Paracelsus lab scene, when Shaton enters the room, the half of the philosopher stone he has drops out of his jacket. But then Paracelsus finds it in his pocket. (00:36:00 - 00:37:00)
Continuity mistake: When Myka's sister hands her the poisoned cup of tea, Myka holds it in her right hand. In the next shot the cup is suddenly in her left hand.
There's Always a Downside - S4-E4
Continuity mistake: When Myka and Jinks first see the trumpet player, the camera stays on him while he plays. At one point he plays a long tone and the camera shows him press the first two valves as if to change a note. The note he is playing continues uninterrupted however. While there are some notes that can be played with more than one fingering, changing valves in the middle of the note always produces a momentary interruption of the note, at the very least.
There's Always a Downside - S4-E4
Continuity mistake: Right at the end when Artie and Pete are talking, Artie is looking down into the bag holding the artifact. The shot changes to his other side and he is instantly looking up and into the distance.
All the Time in the World - S4-E19
Other mistake: Charlotte enters the Duchess' bedroom, walks past Pete & Myka, with her hand behind her back, hiding something. She stands in front of them, trained security people; neither of them notice the gun in her hand (which is visible to the camera, so it is not hidden via artifact), before she shoots her husband.