In two of the Synoptic Gospels it is the arrest and execution of John the Baptist that seems to launch Jesus' ministry. Whenever I read these narratives I do so with an eyes toward Josephus' description of the Baptist in Antiquities 18.5.109-119. There Josephus writes that Herod Antipas executed the Baptist preemptively because he feared that the crowds may do whatever he suggested. In context, it seems that while Josephus won't call the Baptist the leader of a rebellion (nor do the Evangelists) there was some concern on the part of Antipas that this is who he was. The Evangelists focus solely upon the Baptist's rebuke of Antipas' marriage to his half-brother's wife Herodias as the cause for Jesus' arrest. While it is quite possible that this was a motivating factor it seems more likely that the reason given by Josephus would be the central reason for arresting John. That he has an opinion about the lawlessness of Antipas' marriage is nothing if his opinion wasn't dangerous. It makes sense that the Baptist likely critiqued Antipas about not only his marriage, but about many things, and that his criticisms combined with his popularity made Antipas uneasy.
This sheds some light on the intensity of the transition from John to Jesus. While the Fourth Gospel begins this transition while the Baptist is active in his ministry (see Jn. 3:23-30) the Synoptics do not. Mark 1:14 states that it was when the Baptist was arrested that Jesus went into Galilee and began "preaching the Gospel of God." In other words, Antipas' arrest of John would not stop the proclamation that the Kingdom of God was coming. Instead, as the Baptist is removed from the scene Jesus emerges continuing his message.
In Mark 6:14-29 Antipas is presented as fearing Jesus because he thinks Jesus is John redivivus. Antipas has the Baptist beheaded, yet here is John again! Whatever Jesus was doing the reports of those deeds were so that Antipas feared that this was John with new powers (v. 14). In Mark 6:30-44 the people seem stunned and lost now that the Baptist is dead and even as Jesus attempts to move to a secluded place (v. 32) after he hears of the execution the crowds find him. Whatever the Baptist was saying about the Kingdom of God was powerful enough of an idea for the people to refuse to quit now that he was dead. Instead, at least according to the Markan presentation, they sought a successor.
Matthew 14:1-12 follows the Markan script from 6:14-29 with Antipas fearing that Jesus was John resuscitated. Similarly, when Jesus goes to get away from everything after the news reaches him about the execution the crowds find him. Their hope cannot be squashed; they must find a successor. In Matthew 4:12-17 the picture of Mark 1:14 is repeated: John is arrested, so Jesus goes into Galilee. Jesus' message was like that of the Baptist: "Repent, for the Kingdom of Heaven is at hand."
In Luke 3:18-20 the Evangelist summarizes John's rebuke of Antipas and Antipas' choice to arrest him. Then in 4:14-15 he follows Mark 1:14 and Matthew 4:12 by sending Jesus into Galilee after he was tempted in the wilderness, but he doesn't mention John's arrest here. In 9:7-9 he abbreviates the story of Antipas fearing that Jesus was John redivivus, in effect, removing the sense of anticipation found in both the Markan and Matthean versions where Jesus assumes John's role as the voice and hope of the people. One is tempted to think that part of Luke's rational for juxtaposing Jesus and John so early in his Gospel, connecting them as family, might be that he wanted to avoid what Mark and Matthew don't directly suggest, but may imply, that Jesus was John's successor at his arrest.