Even before “Hell’s Kitchen” opened at the Public Theater last fall, it seemed inevitable that a jukebox musical inspired by the life and featuring the music of Alicia Keys, one of today’s most widely admired pop and R&B stars, would transfer to Broadway. Yet in some ways, the downtown venue was a more natural home for “Kitchen,” the story of a scrappy teenage girl’s early coming of age as a woman and an artist. 

The show has nonetheless arrived at the Shubert Theatre, with its excellent principal cast (lovingly directed by Michael Greif) and its buoyant book and score intact — and, for the most part, the transition feels seamless. The company is led by the bubbly Maleah Joi Moon, who made her professional debut in the Public staging as 17-year-old Ali, a character based on the young Ms. Keys, though there are striking differences: At Ali’s age, Ms. Keys had already graduated the Professional Performing Arts School at the top of her class and was securing the record contract that would make her a sensation three years later.