Similarly, Jesus passes the torch to his disciples after raising from the dead, but before ascending into Heaven. The disciples, who have up to this point been mostly incompetent, pull it together and kick-start the early church in a powerful way.
And even before either of those, Moses put Joshua in charge of leading the people of Israel when he was allowed to go no further.
Harry Potter and the Half-Blood Prince, big time, with Dumbledore explaining to Harry about Voldemort's Horcruxes, and including him in the mission to find and destroy them. After Dumbledore's death, Harry and his friends take up the task. Later, in Deathly Hallows, Harry clues in Neville to take his place as he goes off to die.
Neville does this twice, taking over as leader of Dumbledore's Army and the general rebellion against the Death Eaters while Harry wanders around looking for souls to reave.
Used to a heartbreaking effect in the end of The Road.
In Deltora Quest Lief's father has a crippled leg from a tree falling on him so he isn't able to go look for the gems in the Belt of Deltora. He passes the Belt-and the duty-to Lief and Barda the "beggar" who lives outside the forge and spies on the Grey Guards talking about the plans of the Shadow Lord.
Halfway through the Warrior CatsExpanded Universe novel Bluestar's Prophecy, Pinestar, leader of ThunderClan runs off to be a housecat and passes the torch to Sunfall.
Also, at the end of Sunrise Leafpool is forced to give up being medicine cat and Jayfeather becomes medicine cat instead.
Codex Alera has a rather enforced variant of this. 80% of the previous generation of leadership has died at the end of the Vord War, including Gaius Sextus.
Eventually going to (and in the process of) happen in the Star Wars Expanded Universe as the movie heroes die from old age; though the number of people to pass the torch too is running low after several novel series of killing or writing off the new generation.
In Dagger-Star by Elizabeth Vaughan, Red Gloves liberates the land of Palins from a tyrant, and is Offered the Crown, but she just wants to retire in peace and marry her beloved Josiah, plus politics is not her thing. She notes that hardly anyone has actually seen her in person, and that a girl named Gloriana is both knowledgeable in politics and fits the description of The Chosen One like her. Red passes her name and signature gloves to Gloriana, who becomes Queen with most of the people none the wiser. By the time of the book Destiny's Star, Gloriana has become a swordswoman roughly as skilled as Red Gloves.
In The Final Reckoning, the last novel in Robin Jarvis' Deptford Mice trilogy, the Starwife chooses Audrey to be her successor and tricks her into performing a ritual that irrevocably ties her to the position, whether she likes it or not. She doesn't at first, but gradually accepts it as the powers of the Starwife begin to manifest themselves within her.
Kurik's family has served as squires to the Sparhawk family for several generations, so when Kurik is killed by Adus, his son Khalad takes on his father's duties.
Dolmant's rise to the rank of Archprelate upon the death of Cluvonius.
This is not counting Ehlana and Avin, who become Queen of Elenia and King of Thalesia, respectively, upon the death of their respective fathers.
Done between an elderly Bilbo Baggins and his younger cousin Frodo in The Fellowship of the Ring, albeit with a little reluctance. Bilbo, who has retired to Rivendell, at first offers to take up the quest that Frodo ultimately accepts. But it's clear that the old Hobbit's adventuring days are past, and Bilbo passes on his Dwarvish armor and Elvish blade to his successor.
In Parzival, the poem ends with Parzival becoming the new Grail King.
It turns out that the tour of the Wonka Factory in Charlie and the Chocolate Factory is a Secret Test for Willy Wonka to find a good, honest child whom he can train as his successor and thus pass the torch of running his factory to, as he is Older Than They Look and has no family. Depending on the adaptation, the story ends either with Charlie simply becoming Mr. Wonka's protege (as in the book) or this trope taking effect immediately and making Charlie a Grade-School C.E.O..
An essential part of The Giver's relationship with the Receiver. Every Receiver accepts the job with the understanding that he/she will eventually become the next Giver, with the task of passing on the accumulated memories to the next Receiver.
At the end of the first book of the Erebus Sequence, the protagonist declares that he'll be leaving. He puts his friends in charge and says that he may return if people give them too much trouble. The second book, therefore, shifts its focus to one of those friends, with the first protagonist remaining off-stage until the very end.
At the end of Clare Vanderpool's novel Moon over Manifest, Hattie Mae Macke, whose column about local news appears in Manifest's local newspaper, hands it over to Abilene Tucker, who has spent the summer learning about her father's history in Manifest before she was born.