In “Red, White and Blue,” the fifth and final film of Steve McQueen’s Small Axe anthology (and the third to be shown at this year’s New York Film Festival, after the lilting reggae house-party movie “Lovers Rock” and the wrenching social-protest courtroom drama “Mangrove”), Leroy Logan (John Boyega), a British research scientist, figures that he’s had enough of the lonely work of staring at tissue specimens through a microscope, so he decides to become a member of the London Metropolitan Police Force. At his big meet-the-commission job interview, the conversation dances around the issue of race for about a millisecond until Logan puts it right out there, saying that he’s applying for the job “to combat negative attitudes,” and to be a force for change against “divisions” and “misunderstandings.” The crusty officer in charge looks at him and says “You’re right,” and then adds, “Attempts to interact with your people have fallen quite short.”

He’s sincere, and means well, but the problem he’s referring to — the systemic racism of the British police — is encoded in the very phrase “your people.” Taken together, the Small Axe films, which dramatize the lives of West Indian immigrants in London from the late ’60s through the ’80s, add up to a scalding and inspiring vision — a grandly specific panorama of the terror and vibrance, rage and exhilaration, and desperate heroism of a community fighting its way from the fringes of society to the center. The police, time and again, are the battering ram used against them. So what happens when one of the heirs of the disenfranchised sets out to join the police?

What the characters in these films want to do is to live their lives like anyone else. They want to be seen and heard, and to be free of harassment. But they’re also demanding to be viewed not as outsiders who are somehow “accepted” but as the citizens they are. Not as “your” people but, simply, as British people, part of a society that their very presence has made over into a melting pot.

McQueen shows us the gnawing spiritual agony of not having that.

Of the five Small Axe films, “Mangrove” is most definitively a movie (and a fantastic one), and so, in a much smaller way, is “Lovers Rock,” though that one has a semi-wordless flow that makes it more of an infectious curio. Set in the mid-’80s, Yet it’s the only one of the three I’ve seen that feels less like cinema than good TV. McQueen, who wrote the script with Courttia Newland, has crafted the true-life drama of a man who looks like he could turn out to be the Frank Serpico of Black British police officers. And that’s kind of what Leroy Logan was. But this, in essence, is the opening chapter of his story, so it doesn’t quite have the payoff we’re hoping for.

Logan, born and raised in Britain, is drawn to the police because he thinks it would be exciting work. But his decision is also rooted in the experience of his father, Kenneth (Steve Toussaint), a West Indian transplant who’s been hassled by the police so often — all because of who he is rather than anything he’s done — that his rage about them is always percolating at a semi-boil. (In the film, he gets beaten up by two cops when he dares to question an unfair parking violation.) Toussaint, who looks like a feral Eddie Murphy, brings a touch of proud sorrow to his performance as a man who wears every wrong on his sleeve. Leroy, in psychological terms, is joining the police to fix what’s wrong with them and thereby heal his father.

When Leroy first walks into the dorm where he’ll be staying during the six-week training period, he hears Marvin Gaye’s “Got to Give It Up” echoing through the walls (a good sign), and the trainee who’s playing it acts like a stand-up mate. The whole program treats Leroy with respect, and he’s the star student, running relays when everyone else drops and acing the written exams. But when he puts on the full uniform and bobby hat and stares at himself in the mirror, the moment ripples with the fear and authority and skepticism that Boyega holds, all at once, in his eye. He’s staring at the image that terrorized his father, an image that might still think of terrorizing him. And now he is that image. Or is he? Can he change what it means?

“I thought you were cool,” says his old mate Leee (Tyrone Huntley), who happens to be the lead singer of the ’80s British dance-pop trio Imagination. “What happened to you?” He’s kidding, but wherever Leroy goes in his constable guise, he gets no respect. Delinquents pummel him with taunts of “coconut” and “Judas,” he overhears sneering racial putdowns from some of his fellow officers (not all of them, but it doesn’t take many to create an atmosphere of implicit terror), and someone scrawls a horrific slur on his locker. He rises above this stuff, but the larger problem is that there doesn’t seem to be much he can do about it. It’s a locked-in system, a monolith of tribal attitudes. Anyone who goes against it is going to be rejected.

The soundtrack is dominated by Al Green songs like “Tired of Being Alone” and Green’s cover of “How Can You Mend a Broken Heart,” and while they’re effective mood music, the songs came out in the early ’70s and don’t match the period. There’s one moment when you feel the dangerous tingle of London in the mid-’80s: when Leroy is first on the beat, patrolling a neighborhood, and Melle Mel’s “White Lines (Don’t Don’t Do It)” plays in the background, and suddenly the music isn’t a comfort — it’s the hyped heartbeat of a new world.

Yet “Red, White and Blue” mostly lacks the gritty period flavor of the other Small Axe films. It’s a little glossed over. The (minor) daring of the movie is its downbeat narrative. It’s structured like the air seeping out of a tire, so that it presents us with a character of idealistic strength, commitment, and personal heroism only to plop him into a set of circumstances that won’t allow him to be a hero. At the end, when Leroy sits in the kitchen drinking with his father, and they brood over how the wheel of change turns with agonizing slowness, Leroy says that there are times when he wishes they could just scorch the earth and start over. “Red, White and Blue” refuses us the comfort of triumph. It’s caught in the moment we’re in, when hope waltzes with despair and the only thing that’s real is the struggle.

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