The movies, in case you didn’t notice, are having an LBJ moment. It began two years ago, when Tom Wilkinson played President Lyndon B. Johnson in “Selma” as a cagey but courtly political warrior whose hand was forced by Martin Luther King Jr. More recently, the Broadway drama “All the Way” was made into an HBO movie — 20 years ago, it would have been big-screen awards bait — with Bryan Cranston reprising his edgier, closer-to-the-bone performance as LBJ (more cussed, better accent, truly astonishing makeup). Cranston has made a specialty of playing decent men in scurrilous armor, and his LBJ was a cunning vulgarian blowhard saint, with a bark worse than his bite — a cracker-barrel curmudgeon.

The first time you see Woody Harrelson in “LBJ,” a biopic directed by Rob Reiner, you’re aware of how latexy his make-up job is, especially compared to Cranston’s (at moments, he looks like one of the Blue Meanies from “Yellow Submarine”). As a movie, “LBJ” has the Reiner touch — which is to say, it’s square and sincere, like a prestige pic from the ’80s, with gloomy melodramatic “dark” lensing, as if someone were trying to remake “The Godfather” as a movie-of-the-week.

But then, in the opening scene, when Lyndon is still Senate Majority Leader, he chews out his fellow Texan Ralph Yarborough (Bill Pullman) by snapping, “The only thing more annoying than a liberal is a liberal from Texas,” and the thing that makes it a perfect LBJ line as well as a perfect Woody Harrelson line is that he isn’t joking — you can hear the testy, bullying annoyance. Then he asks an underling how many votes he can count on in an upcoming Senate showdown, and when he’s told “about 45,” the answer spins him into a rage. He doesn’t want to know about how many votes. He wants the exact total, and in that moment he becomes a bigger SOB than Cranston and Wilkinson’s LBJs combined. The fury, which leads Lyndon to lightly threaten to cut off the messenger’s member, is compelling, because it’s unhinged but also manipulative (and down-and-dirty funny) in just the way that Johnson could be.

So why LBJ, why now? It’s been half a century since the war over Civil Rights, and “LBJ,” like “Selma” and “All the Way,” tackles how that revolution in the spirit and life and racial-political reality of the United States was, in many ways, incarnated by the evolution undergone in LBJ himself. He was a low-country Southern aristocrat who emerged from a Jim Crow world — and had the world around him not been changing, he would probably have been fine with keeping the status quo. That’s the essential mystery of Johnson. Once he became president, on the day that JFK was assassinated, he committed himself to Civil Rights and carried it through; the Civil Rights Act of 1964 almost certainly wouldn’t have passed had it not been for his strategizing and cajoling and arm-twisting will. But the deeper issue is: Where did that commitment come from? How much of it was “political” and how much of it emerged from who he was and what he believed in? Was he throwing a bone (as some think) or knowingly and righteously bashing down a door?

The reason the question matters is that 50 years later, when America, once again, feels torn and frayed by sins of racial violence, the issue of why it’s necessary to move forward looms large. When you watch a biopic, the hero — whether a musician, a whistleblower, or a president — stands in, on some level, for all of us, and maybe the movies are having an LBJ moment because Johnson, in his tug-of-war between complacency and idealism, racism and brotherly love, reflects something about a great many of us: that spirit inside that makes us want to be better, but only because we know, deep down, that we’re not all we should be. His evolution is ours.

You could say that Reiner comes at “LBJ” from a standard progressive showbiz angle; you can almost hear LBJ saying, “The only thing worse than a Texas liberal is a Hollywood liberal.” But Reiner has been around the block enough times to relish everything in Johnson that’s rudely incorrect. His staging is a little flat, but Joey Hartstone’s script juggles three different time periods, and Reiner makes that work. We see Johnson in 1959 and ’60, when he’s yearning to run for president and finally does (after having assured Robert F. Kennedy, during a private hunting excursion, that he has no such intention), only to realize that it’s not his time, and maybe never will be. His sense of how thoroughly he’s trumped by the Kennedy handsomeness and charisma and Harvard rich-kid glow leaves him with a nearly Nixonian sense of defeat.

Lyndon uses his height and his bellicosity to dominate any situation he’s in. But when JFK makes him the offer to be his running mate, Johnson knows better than anyone that he’s being elevated and neutered at the same time. In “LBJ,” his dynamic with the Kennedys is fascinating. He resents Jack, but also likes and respects him, and Jeffrey Dononvan’s performance as JFK, from his voice to his smile, is slyly convincing. Robert Kennedy (Michael Stahl-David), on the other hand, hates Johnson because he considers him the past, and he does his best to leave him twisting in the wind.

It’s Johnson who realizes that he may be the only one on Capitol Hill who can commune equally with Kennedys and Southern Democrats (“I’m the only one who speaks both languages”). There’s something admirable yet also a little disquieting — mob-like — in the way he puts on two entirely different personalities: progressive Lyndon and good-ol’-boy Lyndon. The wall between the two is knocked down on the day that JFK gets killed. In Dallas, we observe Lyndon up close and see his grief, his burden, his ceaseless savvy. Once he becomes president, the job weighs on him like a curse because of how he got it, yet the cataclysm of JFK’s murder shakes something loose in him.

Harrelson, born and raised (until he was 13) in Texas himself, fuses that distinctive sing-song voice of his with LBJ’s more laidback drawl, and as the movie goes on, he becomes Johnson. He delivers obscenities in a way that’s still shocking. The swearing is part of Lyndon’s cred, his way of signaling that he may be a Southern traditionalist, but he’s no “gentleman,” so don’t even think about pushing him around. He’s going to push you around, and he does that to more or less everyone except his devoted wife, Lady Bird — played by Jennifer Jason Leigh, who looks remarkably like the real thing and nails her homespun resilience.

Harrelson’s Lyndon favors some pretty extreme gallows humor. “What did Abraham Lincoln say when he woke up from a three-day drunk?” he smirks. “I freed the what?” His closest confidant is the Georgia senator Richard Russell Jr. (Richard Jenkins), and here, as in “All the Way,” the drama is rooted in how the Civil Rights battle was tested — and defined — by the fraying relationship between these two long-time bosom buddies. Jenkins gives a daring performance, sprinkling the N-word around with chilling casualness, and it’s through Lyndon’s reaction against Russell that we can see him becoming the leader he is. It’s a transition that’s equal parts realpolitik and intense personal loyalty. Lyndon considers his beloved black cook and housekeeper “family,” and he’s appalled by the cruelty she had to put up with on a simple car trip through the South.

“LBJ” is a prosaically engrossing biopic that lacks the artistry and excitement of a movie like “Lincoln” (whose structure it echoes) or “Selma.” Compared to those films, it will likely struggle to find an audience. Its defining character may be someone who isn’t even in it: Martin Luther King Jr. Yet despite that omission, the film’s (modest) strength is the way it takes the audience up close to the psychological truth of LBJ’s complicated fusion of racism and racial enlightenment. His commitment to the Civil Rights Act becomes a way of seeing where the winds are blowing, but also letting those winds sculpt the sands of his humanity. To change the country, he changes himself.