Jeff Bridges, in his new series “The Old Man,” is playing the kind of spy they don’t make anymore.

His Dan Chase is, even at an advanced age, a master tactician; he repeatedly bests men many years younger than he is in hand-to-hand combat. He’s also winningly persuasive, so much so that the FBI counterintelligence chief tasked with finding him (John Lithgow) can’t resist giving him a head start.

Indeed, the aptly named Chase seems perpetually one step ahead in evading pursuit, and the show is similarly ahead of us at home: We find out just what has made Chase, and what makes the chase, as we go. While part of the CIA, Chase went rogue; he’s spent the rest of his life — before and after becoming a parent and then a widower — living under assumed identities and staying just out-of-reach of an angry defense apparatus. Given the opportunity to simply disappear and sever contact with his daughter (whom we meet as an unidentified voice at the other end of the phone), he refuses.

All of which might call to mind elements of FX’s late, great “The Americans,” with its kicky disguise fantasies, its knotty parent-child relations, and the perverse thrill of Matthew Rhys’ KGB plant Philip developing an affinity for the enemy. But while “The Americans,” in its early going, had the thrill of discovering new situations and new facets of one’s partner, “The Old Man” assays a different stage of life, and comes by its moroseness honestly.

That doesn’t mean the tone doesn’t get punishing. There’s a certain windiness to some of the reflections about aging and about tradecraft, as if Cliché were a language in which one achieves fluency at Quantico. (“It’s not for the faint of heart,” Lithgow’s Harold Harper says of his pursuit of Chase, “but it’s a hell of a drug.”) And while Chase kidnapping a woman to bring her with him on his race to safety was a device whose purpose we can appreciate — merging one last mission with one last chance at new affection — both the way he meets Amy Brenneman’s Zoe McDonald and the way they interact after they hit the road strains credulity. It’s also an avenue for conversations that feel, too often, like unvarnished info -dumps.

The actors do their best to sell it all. Brenneman brings the same sorrowful ache she lent “The Leftovers,” while Lithgow is predictably excellent, in spite of some of his dialogue. And Bridges lends grain and texture to a role that could too easily be a geriatric Ethan Hunt: Chase may be close to infallible, but he also gets meaningfully frustrated and weary. It’s not hard to understand why he’s feeling elegiac. But the show is too breathless by half in making sure you know how and why he’s feeling low; it’s at its best, oddly enough, when Chase slows down.

“The Old Man” premieres Thursday, June 16 at 10 p.m. ET/PT.