Elijah has been mentoring Elisha and he knows he is about to leave this earth. They are now going to pass by what are important places in the history of Israel. In each place, Elisha is asked by Elijah to stay behind while he continues on but Elisha refuses to leave his side. I'm not sure that even Elijah knew just exactly how this day was going to play out. He only knew it was his final day here on earth. Perhaps he wanted to spare Elisha from the sorrow of seeing him die. To me the lesson is if I should be somewhere in the will of God, it is important that I am there. We have an expression that we 'need to be where the action is'. When it is God's action, it is more important yet.
God had obviously revealed to many of these young prophets that this was to be Elijah's final day on earth (verses 3 & 5), but they did nothing about it. (Though in verse 7 we do find that 50 of the sons of the prophets went and stood to view what would happen from a distance.) Mere knowledge doesn't work. I would think something like this should have set them off enough to have wanted to be more than bystanders. I am afraid there are too many bystanders in the church today.
Elisha asked for a double portion of Elijah's spirit. If he had stopped anywhere along the way he wouldn't have gotten it.