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.

Ali, in contrast, is more focused on the strapping young man who plays drums on the corner near the building where she lives with her single mother, a haven for artists in need of affordable housing. “Kitchen” seems to be, as much as anything else, a love letter to Ms. Keys’s own single mom — represented here as a hard-working, fiercely protective woman called Jersey.

We also meet Ali’s father, Davis, a charming musician who drifts in and out of the picture, seldom showing up when he is needed. He and Jersey are respectively played by the redoubtable singing actors Brandon Victor Dixon and Shoshana Bean. Mr. Dixon’s lustrous voice, in particular, is well-suited to the soulful tunes written by Ms. Keys and a bevy of collaborators. (Her numerous co-writers are only listed, alas, in the back of the playbill.) 

In contrast, Ms. Bean, while extremely likable in her role, can try a little too hard. A Broadway belter by nature — previous roles include Elphaba in “Wicked” and, in regional, Fanny Brice in “Funny Girl” — the actress can sink too easily into vocal melodrama, which with these songs often entails the kind of hyperactive embellishment that pop singers have regularly indulged in since Mariah Carey’s early heyday.  

At a recent preview, delivering one song in which Jersey expresses her frustration at Davis, Ms. Bean had what might be best described as a melisma meltdown, bending and shouting the notes as if she were trying to stomp them out. In fairness, the crowd loved it; there’s no easier way to win a contemporary Broadway audience’s heart than to sing loudly and with lots of flash.  

Ms. Moon’s own sterling vocals seemed rather too heavily amplified at the beginning of this performance, but Gareth Owen’s sound design generally serves the numbers, which include a sprinkling of new songs, well. The same can be said for Camille A Brown’s choreography, which while occasionally overzealous captures the aspirational energy of Ms. Keys’s work.

Kristoffer Diaz’s book, like others crafted for jukebox outings, can be too cute or obvious in weaving in that work. As Ali’s friends teasingly point out that her eventual love interest, Knuck, hasn’t even noticed her yet, one breaks into “You Don’t Know My Name.” Later, when a meeting between Jersey and Davis to discuss their daughter’s issues leads to flirtation, they segue into “Fallin’.”

A few setups feel less contrived and more compelling: Miss Liza Jane, the older woman who stirs Ali’s interest in piano — played by a poignant, authoritative Kecia Lewis — consoles and motivates the young woman in a defeated moment with “Perfect Way to Die,” a pining account of grief and resilience. And “Un-thinkable (I’m Ready)” becomes a sweet showcase for the budding romance between Ali and Chris Lee’s sensitive, endearing Knuck.

The show closes, as it must, with the enduringly inescapable “Empire State of Mind,” rendered here as a delirious production number that’s at once cheesy and transporting. It’s a vehicle that was built for Broadway, and I suspect it will stick around there for a while.