Van from GUN×SWORD is this, both pre-series and during.
Kenshiro of Fist of the North Star fits this trope to a tee, especially at the beginning of the series. He wanders the post-apocalyptic landscape from town to town looking for his kidnapped lover, has his sidekick in Bat, is theoretically not looking for trouble, and yet somehow always leaves behind body counts that range from dozens to hundreds.
The protagonist, Ginko, from Mushishi (a sort of mononoke-ologist)
Dr. Tenma from Monster (Subverted as any troubles are almost always connected with the "monster" Johan).
Raven Tengu Kabuto, from the anime of the same name. The above description is almost a plot synopsis.
Vash The Stampede of Trigun has this as his whole thing, except he's also a wanted outlaw and an immortal superhuman.
Since his setting is Western-inspired and he has mad gun skills, Chronic Hero Syndrome and multiple issues with settling down, he does a lot of this, especially in the anime which put off the Cerebus Syndrome a lot longer, although most of the story is consumed by the plot happening.
He only stops the drifting thing during his 10-Minute Retirement after he inadvertently shot the moon, when he grows a Beard of Sorrow, changes his name, and doesn't kick the asses of the creeps who come to his new hometown and start killing everybody who annoys them. Which is kind of extreme restraint, given the town couldn't build enough coffins to house all their dead.
Interestingly, that situation gets resolved by Wolfwood drifting into town looking for him the same day Vash's adopted family member is kidnapped, and then both of them kick ass and walk off into the sunset. Vash's But Now I Must Go costs him a lot more than usual this time.
Ran from Kazemakase Tsukikage Ran. ("Kazemakaze" translates into something like "Carried by the wind", a colorful way of describing a carefree wanderer.)
Rurouni Kenshin: The titular character purposefully became this after the war and settled down starting on the first episode. We see him comment that with the friends he makes, he may stop wandering. Also, "Rurouni" can be translated as wanderer. So Yeah.
After being defeated by Kenshin, Seta Soujirou decides to follow Kenshin's example and go wandering, giving himself time and freedom to resolve the moral conflict he's now facing.
Except his dad. That was his hometown, though he hadn't been there in over a decade.
Amusingly, the fallout from this adventure later causes him to need to flee Japan, so given his temperament he probably goes around being this in countries where no one can understand a word he says. Everyone understands when you punch a wall and the house shatters, though.
While he's mostly a Wise Prince, Ashitaka from Princess Mononoke has shades of the Drifter, as he becomes entwined with and takes a stake in the outcome of the conflict between the various factions he encounters during his travels.
As Eboshi says when asked if she saw him come around: "Came, and went."
Kanbe in Samurai 7 shows signs of being this, although the town actively recruits him and he puts up a lot of resistance.
The titular Kino of Kino's Journey, travelling the world on her talking motorcycle, with a strict rule to never stay in one country for longer than three days.
In Sekirei, Mutsu had shades of this in his backstory. After leaving the Discipline Squad, he took to Walking the Earth and only came to the capital after beginning to react to his Ashikabi. Though he complained about not wanting to cause trouble, he intervened in a violent mugging and rescued Mikogami. Any efforts to avoid taking part in the Sekirei Plan were dashed, since his master was Desperately Looking for a Purpose in Life.
Naturally, it happens in Preacher. Jesse Custer indulges in this trope when he drifts into the town of Salvation. Though to be fair, by the time he ends up in Salvation, he had a lot on his mind, not all of which he can recall.
The comic books spun off of the original The Legend of Zelda give this sort of backstory to Link, who happens to wander into Hyrule from his native Calatia just as Ganon is starting to wreak havoc.
The heroes in Sin City have the demeanor of the drifter (quiet loners with troubled pasts), even if they tend to stick to the city limits of Basin. Wallace might be an aversion since his story seems to indicate that he is relatively new in town. He doesn't seem to grasp how corrupt the city is and despite his deadliness, he is a relative unknown.
The only real clue of his motives come from a strange letter he gets in the first chapter from his unseen guardian angel, wherein Jericho is told to travel to a certain town and, in very uncertain terms, kill some people. That's it.
John J. Macreedy in the classic film Bad Day at Black Rock, although he arrives in the titular town with a specific purpose in mind (which is not to clean the place up.)
The first paragraph describes fairly accurately John Rambo in First Blood, right down to being harassed by the law. Only his subsequent actions are not to help the town at all...
Max from the Mad Max films. Shane in black leather.
High Plains Drifter plays with the conventions of this trope a bit, mainly in that The Stranger (as he is credited) is hinted at being the ghost of a man murdered by the townsfolk (indirectly) years prior and thus brings on a little vengeance by turning the town's folk against each other, manipulating and scaring them into giving him absolute power and pretty much ruining the town's economy (by blowing up the hotel, tearing down the barn and not paying for any of the many goods and services he takes advantage of, such as buying everyone in town a drink from the bar at the bartender's expense. In fact, he's barely in the town for ten minutes before he kills three men, drags a woman to the barn and rapes her.
Sanjuro, in Kurosawa's Yojimbo (the prototype for For a Fistful of Dollars) and Sanjuro.
Also Zatoichi, hero of a long-running series of Japanese films. In each film, he wanders into a new Adventure Town, where he at first pretends to be a simple itinerant masseur and gambler. But when some local yakuza boss or corrupt official threatens him or the group of innocent commoners he's befriended, he reveals himself to be a master swordsman and all-out badass. Oh, and he's blind, too.
The animated film Kung Fu Panda starts off with a dream sequence where Po fits this trope PERFECTLY. Of course, this is just his dream self, but it does seem to show how Genre Savvy he is. Two tropes for the price of one?
Sartana is a subversion; while he does go from town to town for adventure, he is definitely not the silent, brooding type.
Comrade Sukhov from White Sun of the Desert. A retired soldier who just wants to go home, he walks the sands of Turkestan and gets into trouble.
In The Man from Earth, John Oldman is moving away, which he says he does every decade. He reveals that he has been doing this for 14,000 years, because people start to realize that he doesn't age.
The hero of Three Iron gets along by breaking into homes which are empty and temporarily unoccupied. He does not steal from the homes like any typical run-of-the mill burglar. In a bizarre, parasitic manner, he lives off the food resources and physical amenities provided by the homes, yet leaves them in a better state than when he first broke in, repairing damaged appliances and washing dirtied clothing thrown around by its owners.
Jack Reacher, in the novels by Lee Childs. Each book starts with him drifting onto an area and ends with him on his way to somewhere else.
The Rangers of the North are very much this trope. They have few if any permanent settlements, and most of the men's time is spent wandering the wilds of Eriador patrolling for servants of the Enemy and safeguarding the villages and roads, even though the people they are protecting regard them with suspicion and fear.
In The Stand, Randall "The Walkin' Dude" Flagg was the embodiment of this trope's Evil Counterpart for a long while, until he decides to settle down and rebuild America in his own image using newfound magical powers. He takes on shades of this trope again when he goes traveling through time and the multiverse, and keeps "becoming", amongst other things, a marine, a member of the KKK, a headsman, a Viet Cong, a supporter of a radical blacks' rights group, a court magician (several times), and other roles that helps his subversive schemes. Really, the only thing that makes him into Nyarlathotep, and separates him from being an evil Time Lord is magic and a lack of a Tardis.
Eiji Hino is only said to be a drifter before Kamen Rider OOO starts, since he stays in the same general area for most of the series, but he goes back to that lifestyle in the end - only now he has a group of friends to stay in touch with.
Before either of them, we had Yuusuke Godai, who like his partial Expy Eiji only stays in the same general area over the course of the series because of the circumstances regarding the Gurongi and his own sense of responsibility as the one bearing the power of Kuuga.
Jared from The Pretender is on the run from an evil organization that held him prisoner from childhood, but finds time to research and uncover injustice and save people's lives wherever he goes.
In Supernatural; The Winchesters and most other hunters as well. The Winchesters have less of a home base than the majority of hunters, and are unusually kind and personable...even if they are violent maniacs with no respect for the law who lie like they breathe.
It might take a bit longer than other examples, but Immortals in Highlander: The Series have to move around every decade or so when people begin to notice that they don't age.
Modest Mouse's "The World At Large" even goes so far as to mention that the narrator relates to songs and books written about drifters.
This is a typical trait for Prometheans. If they settle in any one place for too long, Disquiet starts to take hold in the townspeople and Wastelands bloom up under their feet. Hence, they're constantly on the move, only staying long enough in any one town to enjoy contact or refresh supplies without polluting the land or warping people's minds.
One of the playable archetypes in Feng Shui is The Drifter. He even has the ability to show up exactly where and when he is needed. In game terms, he announces he wants to show up, and everyone picks a reason how he got there. He picks the one he likes the best.
In Dungeons & Dragons, such characters are too often played to the point they have become cliche, earning the pejorative murderhobo. Treating them as what they are (i.e., vagrant serial killers) is a DM's delight.
Perhaps in name only, but in BIONICLE, all six Piraka seem to have nicknames, and Thok's is "the drifter."
Bartz from Final Fantasy V starts off as one of these, with his pet Chocobo, Boco. This was actually his deceased father's last request.
Shadow from Final Fantasy VI actually has a chance to just take off and leave your party after every battle.
Nearly every Fallout game has you playing some variant on this character type... unless you prefer evil karma, in which case you become a Doom Magnet.
Similarly to Fallout, The Elder Scrolls IV: Oblivion allows you to wander the countryside, picking up whatever quests you want, joining any/all of the five main factions, or whatever else you can think of that has nothing to do with the main story.
In Skyrim, the Dragonborn wanders the countryside inbetween/on-the-way-to/during battles with Dragons.
Shiren the Wanderer fits this well. All of the Wanderers (not just Shiren) are always on the move. In fact, its a gameplay feature: if the player stays too long in a single floor they hear a gust of wind, telling them to move on or suffer a Nonstandard Game Over. Also, backtracking is rarely a good idea, because no items spawn in the level and you end up fighting more monsters. It's all there to ensure that you always keep on the move...
Medoute in Blaze Union. She winds up settling down with the rest of the party for a while after realizing she needs to take responsibility for influencing some of their important decisions, but towards the end of the game she remembers that the whole point of her journeying was because she didn't want to have to deal with responsibility and starts chafing. In most of the endings of the game, she leaves and goes back to Walking the Earth.
You're pretty much this way in the Rune Factory games even with your monsters and friends and neighbors. But when you want to go beat the snot out of wild monsters, who helps you out? No one.
Subverted during the first part of Tides of Destiny because Sonja is in Aden's body, you can hear her adding comments when you're fighting and also the changes of the day.