The Influence Character function can be handed off successfully from one Objective Character to another, but it is tricky. There is a section in the theory book on "hand offs" and it covers this topic pretty well. The idea is that the Influence Character function has to be felt throughout the entire story, whether they are actually present or not. The Influence Character is a presence whose impact is felt by the Main Character, forcing the Main Character to face their personal problems. This function can be held in one player and then picked up by another, but the same appreciations have to be at work in both players when they are being the Influence Character; i.e. the same Concern, Issue, Problem, Solution, Critical Flaw, Benchmark, etc.. If two characters in your story carry this function, then they should never meet in the same scene because it will feel like you have two of the same character in there. In a hand off, it is probably best to have the original Influence Character drop back to be less important to the story when a new player becomes the Influence Character. Maybe the original IC should drop out of the story altogether, it's up to you. But the more they hang around after giving up their original function, the more potential for confusion there will be.
The best hand off I've noticed yet is done in Clint Eastwood's In the Line of Fire. The Influence Character function is first held in Renee Russo's character, the woman agent who eventually becomes Clint's partner. But when Clint's first partner is murdered by John Malkovich's character, then the John Malkovich character takes over the Influence Character position. At this point, Renee Russo becomes pretty much an archetypal sidekick. The thrilling storytelling at the time of this switch helps hide what's really happening. The author's also seemed to really have a firm grasp of how they wanted this to work, so they never violated the hand off and successfully had two characters represent the Influence Character function.
Your question makes me think of another example of how an Influence Character can be woven into a story in an unconventional way. The play The Glass Menagerie by Tennessee Williams has an Influence Character who doesn't actually appear on stage to say any lines until the last third of the play. The Main Character in this play is Laura, the meek daughter who is kind of hidden in the play by her lack of dialogue and activity. But her devotion to an unrequited love from her old high school is brought up regularly in the play, and this person is coincidentally invited over for dinner toward the end of the play. This gentleman caller is the Influence Character, and the final scenes allow Jim O'Conner to continue his role as the Influence Character in person. This example illustrates how the Influence Character has to be present throughout the whole play in some manner or other (like in Laura's little shrine to Jim), but does NOT have to actually be there in person for every single act.