What are the advantages to writing a narrative that shifts back and forth in time for an author?

I do not think you are talking about flashbacks, but the entire structure of a book. Off the top of my head the joy luck club comes to mind, which circled around a group of daughters and their mothers, the book alternated sections where they told the mothers stories of the past, the daughters stories of the present, and also the mothers views of the daughters when they were kids. It also had flashbacks in the present. This worked well for the author trying to tell a story of family, understanding and how family can cause a past that isn't directly ours to effect who we are.

If your story is centered around one character, and it is shifting back and forth in life for her/him, their can be many benefits, however, I have not read many books that do this. "The Pale King" by DFW is one, however, there are so many devices employed in that book it would take time to dissect that particular ones effect on the narrative.

I will give what I think is the main benefit of having a single plotted novel that shifts back and forth in time: a radically different, hence some may say better, understanding of the character.

If this is something you wish to pursue, I would advise you to see the movie "La Vie in Rose" about the life of Edith Piaf. The movie tells the story of a singular narrative, sort of about her life and more about her being an artist. It does this by jumping all over the place. Not just back and forth but combings storylines in ways that present ideas, showing moments and then shifting somewhere else leaving the moments unresolved till far later in the movie. You really have to see the movie two or three times to realize kind of how they pulled it off. But by the end you feel like you know her more than if you had of just watched a beginning to end story, even by showing in early parts of the movie how she was older, you understand the child part more because you saw the fortitude she used to overcome that.

I hope this has been helpful. I too am working on a memoir in novel form that has no disenable linear narrative as of yet, because the main focus of the story is internal, and whatever events it can use to promote that story in a linear way, it doesn't matter when they happened.