Continuity mistake: When Castle gets back from helping Alexis with her friend and thanks Beckett for taking care of Fagan (Alexis' egg), just after her line, "Yeah well, he was easy. He didn't even fuss when I put him to bed", Ryan and Esposito come in and Ryan is holding a missing persons poster of Elizabeth McGinty. As soon as they walk in, Ryan is holding the poster with the front facing the camera. But two shots later he is holding it with the back facing the camera.
Audio problem: When Beckett is talking to Matthew's family about his "personal ghost," as his sister put it. About half-way through her line, "I showed some of Matthew's drawings to one of our therapists and he said it looked like he might have witnessed something disturbing, seen something very violent", there is a shot from behind Beckett facing Matthew's parents. In this shot, Beckett's lips don't match what she is saying.
Answer: The networks are now having winter breaks. A mid season end of new episodes, when they come back after the new year, it picks up as if several months have gone by. Beckett suggested Valentine's Day as a "chick flick" to watch as she is moved in. Also some episodes are not scheduled to air in order.