Edmund, Lucy, and Eustace are pulled from their own world through a magical painting into the eastern seas of the world of Narnia.
The call of the heroes in this book is difficult to miss: our three protagonists are literally pulled into the world of a painting on the wall. There's clearly specific magic at work, targeting them and making sure they arrive at the correct place and time to join King Caspian's eastern voyage. As far as we know, the painting they get sucked into has never come to life for anybody else, and they're plopped down right next to the Dawn Treader just as it's left familiar territory and is about to undergo its first adventure.
King Caspian and his friends travel east in the Dawn Treader seeking the seven missing lords and the end of the world.
Although this journey takes place over the course of several different adventures, each with its own corresponding island, several things are consistent throughout that indicate that this is really one single stage. On each island they visit, Caspian and his friends find traces of the missing Narnian lords – sometimes information, sometimes belongings they left behind, and occasionally the lords themselves. Each island visited also marks another step in the journey eastward, so that the islands are like mile-markers showing their progress.
Arrival and Frustration
The Dawn Treader sails as far east as it can, stopping just before it runs aground. Caspian's attempt to abandon his country and continue on the adventure is thwarted by his friends, crew, and Aslan.
Although King Caspian gets close to the utter east, he doesn't actually get to see the wall of water flowing upward that marks the easternmost boundary of the flat world in which Narnia exists. That's a privilege reserved for Reepicheep and the three visitors from "our" world, Edmund, Lucy, and Eustace. Caspian's responsibilities as ruler of Narnia, and his developing relationship with Ramandu's daughter, require that he turn back and allow others to complete the quest for him.
The Final Ordeals
Caspian must return to Ramandu's island and to his duties as King of Narnia. Edmund, Lucy, and Eustace are sent back to their own world through a door in the sky. Reepicheep sails into Aslan's country.
In The Voyage of the Dawn Treader, the "final ordeals" that most of our characters must go through consist of leaving the adventure behind and going back to their regular lives. For Edmund and Lucy, the ordeal involves the knowledge that they will never get to come back to Narnia; they must learn to recognize Aslan in another guise in their own world. For Caspian, the "ordeal" is the duty he must fulfill: going back to rule Narnia. Not that we think that will be all that unpleasant, but it will involve a lot of day-to-day problems, struggles, and politics. The only adventurer who really gets to go all the way is Reepicheep. Presumably he, too, must go through some "final ordeals" before he gets into Aslan's country, but we're not told what they are.