The Bequeathing is a trial in which a Keyblade wielder on the level of a Master chooses a person to inherit the power to wield the Keyblade. If the Keyblade accepts the inheritor selected, that person will become able to summon their own Keyblade whenever attempted.
A Bequeathing ceremony may be either intentional or accidental. To pass down the Keyblade intentionally, a Keyblade master must choose a successor. All the master must do is test the strength of the one they have chosen by allowing him or her to hold their Keyblade. The Keyblade is accidentally inherited when one of a strong heart and will makes contact with it.
Kingdom Hearts Birth by Sleep
Some time before the Mark of Mastery exam, when Terra, Ventus, and Aqua were still training together, Ventus felt down about losing to Terra in a mock battle. Terra cheered him up, performing a mock Bequeathing, in which he recited the mantra and passed his Wooden Keyblade on to Ventus.
The Bequeathing is unintentionally performed when Aqua visits Radiant Garden and comes across a young Kairi, who was being attacked by a swarm of Floods. Kairi runs to hide behind Aqua and touches the Keyblade, which causes her to become an inheritor.
Aqua also had the intention of performing the ritual with Sora when she met him in Destiny Islands, but she noticed that Terra had already made it with Riku. Not wanting them to fight like she and Terra would, she refrained, although Sora and Riku would inevitably clash with each other several times ten years later.
The Bequeathing is intentionally and correctly performed by Terra when he meets a young Riku on the Destiny Islands. Seeing himself through the boy, someone who would do anything to protect his friends, Terra proceeds to select Riku as his successor. Many years later, this showed that Sora was not originally the Keyblade's chosen wielder when he received it. However, due to Riku's descent into darkness in Kingdom Hearts, the Keyblade was led to choose Sora, as Ventus's heart, the heart of a Keyblade bearer, resided within him. Riku would not regain the Keyblade until one year later on his path to return to the light.