This passing of the torch has become a common occurrence, with new Bonds being appointed.
The audition process is always the same, as they want actors who can encapsulate in one scene all the elements that make Bond tick with audiences–that is, a genuine yet iconic James Bond.
Producer Michael G. Wilson revealed in a conversation at the BFI that all potential Bonds audition with the same exact scene from From Russia, With Love, the second Bond picture (after Dr. No)
“From Russia, With Love” was the second James Bond film, but is possibly the most iconic. The story of Bond helping a Soviet defector while the evil group SPECTRE attempts to assassinate him is a simple entry into the franchise, but it allowed Connery to do all the iconic James Bond things. He drinks martinis, he shoots guys, and he gets beautiful women. It makes sense then that this film is used for the audition, as it’s really Bond at his basest form.