“Logan,” the third and final chapter in the “Wolverine” series (though it’s the 10th — count ’em — “X-Men” film, and not necessarily the last), opens in 2029 on the outskirts of El Paso, where Logan (Hugh Jackman) steps out of the beat-up limo he now drives for a living to confront the goons who are trying to strip the car. To say that Logan looks the worse for wear would be putting it mildly. He’s a drunk, with mottled skin and his trademark sci-fi muttonchops grown into a scraggly beard. (The beard looks like a virus that’s going around, as if Jackman were warming up to play Mel Gibson.)

Optics aside, it’s Logan’s mutant prowess that’s really been tamped down. In any previous installment, a skirmish with mere mortals wouldn’t have been much of a contest, but Logan’s razory knuckle-blades no longer pop with the same alacrity, and though bullets don’t kill him, he doesn’t quite bounce back from them either. (Later that night, he has to force them out of his chest.) Is Wolverine growing old? Yes, but it’s worse than that. After all these years, his admantium enhancements are poisoning him, a process that can be delayed with the right serum but not reversed. Then again, it would be understandable if he were also feeling a touch of “X-Men” fatigue.

As a movie, “Logan” takes a cue from its hero’s slowed-down metabolism of invincibility. Directed by James Mangold, whose last feature was “The Wolverine” (2013) — one of the most dynamic entries in the “X-Men” cosmos — the new film doesn’t try to be a shoot-the-works, how-crazy-are-his-powers grand finale. It’s a scruffy dystopian road Western that takes its time in a way that most slam-bang superhero movies don’t. And the analog pace and elemental story work for it. Each time the violence explodes, it’s slashingly satisfying, because it’s earned, and also because Mangold knows just how to stage it.

“Logan” doesn’t get lost in CGI overkill or annoyingly messy Tinker-Toy franchise plotting. It’s a wholehearted drama made with a shot language that looks nearly classical. It must be said, however, that the story often feels stitched together from other films, a quality made explicit when the characters watch an extended scene from “Shane” on TV. “Logan” isn’t as darkly exciting as “The Wolverine” was. With its hero suggesting a broken-down cousin to Mad Max, it’s like “The Road Warrior” meets “Shane” meets “Terminator 2: Judgment Day” (yes, there’s a “bad” Wolverine). But that turns out to be a recipe that brings the saga to a satisfying close. Just about every fan of the “Wolverine” series is likely to feel well-served, and you can do the box office math from there.

The best thing about “Logan” is that it’s one of those movies about a grown-up killer who becomes the mentor and protector of a child, yet it manages not to be cloying. The kid, in this case, is 11-year-old Laura (Dafne Keen), a dark-eyed urchin of silent ferocity who comes under Logan’s wing (or maybe I should say his blade-claw). Wolverine, we’re told, is one of the only mutants left. In “Logan,” they’ve faded away and become cultural relics, and that’s one of the sources of Logan’s weariness. He keeps Charles Xavier (Patrick Stewart), now 90, hidden on a rusted-out farm below the Mexican border, where Professor X is subject to brain seizures that paralyze everyone around him by making the air molecules pulsate with menace. But then Laura shows up, dumped into Logan’s life by a Mexican nurse (Elizabeth Rodriguez) from a local clinic. She’s a mysterious girl, who says nothing but carries herself with a confidence that’s unearthly. She’s like a version of the Feral Kid from “The Road Warrior.” You could also say that she’s a chip off the old blade.

Keen, in her movie debut, has the orbs of a staring bird and an air of preternatural awareness. She could be the junior sister of Rooney and Kate Mara, and that’s because she holds the screen with her solemnity. Logan agrees to drive her to Eden, a utopian refuge for mutants in North Dakota — though, as he discovers (in one of the film’s few funny gambits), Eden originated in the “X-Men” comics, which in Logan’s mind means that it has to be a made-up place. For most of the movie, he, Laura, and Professor X are on the run from Dr. Zander Rice (Richard E. Grant) and his goons. That’s the whole plot, but Mangold strikes a nice balance between road-movie ambling and eruptions of feral suspense.

It’s Jackman who holds “Logan” together and gives the film its glimmer of soul. He has been playing this role, more or less nonstop, for 18 years, but he seems startlingly not bored by it. Better still, he’s a more refined actor now than when he started, and in “Logan,” he gets to play something rare in comic-book cinema: a powerhouse of animal rage who is slowly, agonizingly slipping away. By the end of the movie, he gets his muttonchops back and reminds you, once more, of what’s great about this character — his hellbent quality, embodied in those flesh-ripping kills that are his way of making good on a mutant destiny he never asked for. No “X-Men” movie will ever be great (the material is too derivative), but Jackman, though he’s the Superman of the bunch, has gone deeper into the alienation than any other mutant in the series. The end of “Logan” is genuinely touching, as Jackman lets you feel the character’s strength and pain, and — finally — his release.