The only sources that we have for the movements of Jesus are the gospels, so any reliable source must be based on these. However, reliably tracking the movements of Jesus is fraught due to differences among the four canonical gospels. These differences arise out of the sources that each evangelist used, as well as the interpretation that each evangelist placed on that material.
Bear in mind that all four New Testament gospels were originally anonymous, and that most critical scholars say that none of the gospels could have been written by an eyewitness to the events portrayed. Whenever a difference exists, scholars must decide which account is more likely to be what really happened. For this, they need to understand the history of the gospels. John Dominic Crossan says in The Birth of Christianity, page 109, the theory that the Gospels of Matthew and Luke were actually based on Mark's Gospel is held today by a fairly massive (but by no means total) consensus of contemporary critical scholarship. In page 97, Crossan says If Matthew used Mark, it does not follow that Mark gives you the historical Jesus. It does follow that, in those places where he is copying Mark, Matthew gives the historical Mark and not the historical Jesus.
Matthew 12:9-13:54
In Matthew 12:9, Jesus was in the synagogue, apparently close by the field where the disciples picked grain on the Sabbath but had left there, with his location unstated. As his relatives stood "outside", Jesus was once again in a house when his mother and brothers arrived (12:46), although at the same time he was talking to crowds of people. In verse 13:1, Jesus went out of the house down to the Sea of Galilee. We do not know where the house was, and are not told when or how he arrived there, although it was close by the Sea. He got into a boat so that he could address the crowd lining the shore.
Then, dismissing the crowds (13:36), he went back to the house. When Jesus finished explaining his parables to the disciples, he went away from there and to his native place and taught the people in their synagogue (13:53-54). His "native place" would be the town of Nazareth in Galilee. There is a break of unknown duration in the narrative, as we hear about the fate of John the Baptist (14:1-12), then we return to Jesus who went "from there" by ship into a desert place.
Mark 3:20-4:35
In Mark's Gospel, Jesus is already in his home town (Mark 3:20) when he taught two of these sayings, then his mother and brothers came (Mark 3:31-33). On another occasion (4:1) he began to teach by the sea. This corresponds to Matthew 13:1, except that there is no suggestion that the teaching by the sea is on the same day as the arrival of Jesus' mother and brothers (cf Matthew) - necessitated by the distance from Nazareth to the sea. On the evening of the same day (4:35), Jesus suggested they cross over in this boat to the other side, leading to the event of calming the sea.
Luke 8:1-23
In Luke's Gospel, Jesus was travelling from town to town (Luke 8:1,4) and his mother and brothers came (8:20). There is no mention of the event when he stood in the boat preaching to those on the shore,nor of Jesus preaching in his home town, as we see in Matthew and Mark. It can be assumed the author simply chose to leave these out, but it creates difficulties in trying to identify Jesus' movements. Next, "one day" he got into a boat with his disciples and said to them, "Let us cross to the other side of the lake." So they set sail ...(8:22-23). This is followed by Jesus calming the storm.
Conclusion
We can see by this example, prompted by the question, it is not possible to arrive at a consensus from more than one gospel as to the movements of Jesus.
