Chase Fight

An X Meets Y cross between a Chase Scene and a Fight Scene. At least two combatants are battling in a normal way, but their fight takes them on a journey as well. This could be because one or both fighters is trying to get to a specific destination, or maybe because they're just trying to get away from the beatdown they might be taking.

When on the ground, this could feature copious amounts of Le Parkour and Roof Hopping, these fights tend to go anywhere and everywere. This is also highly improbable, as the fighters, if human, would likely be tired out of their minds to fight, run and give snappy quips at the same time. Of course, the fighters could be simply superhuman.

If this is seen as a video game mechanic in a Fighting Game, that is Free Floor Fighting. See also Get Back Here Boss.

Anime and Manga

 * In Soul Eater this is how the fight between Sid and Mifune goes (while essentially snowboarding without the boards no less).
 * In the Davy Back Fight arc of One Piece, Zoro and Sanji do this as they attempt to reach Tonjit so they can tag him.
 * In Naruto Pain and the demon fox engage in this.

Film

 * The fight between Jack, Will and Norrington from the second Pirates of the Caribbean.
 * Morpheus and Trinity's highway battle against The Twins in The Matrix Reloaded while trying to reach an exit so they can escape the Matrix.
 * The final fights in some of the Star Wars movies tend to be this, for example the Obi-Wan & Qui-Gon vs. Darth Maul fight and the Obi-Wan vs. Anakin fight.
 * G.I. Joe the Rise of Cobra. The scene where the Baroness and Storm Shadow are driving to the Eiffel Tower in order to attack it and fighting with the Joes as they do so.
 * Raiders of the Lost Ark. The running battle between Indiana and the Nazi convoy that's trying to get the Ark to Cairo. Indiana chases after it on a horse, eventually boards the truck and fights everyone involved.
 * The Blues Brothers. The title characters are trying to get to the Cook County Assessor's office to pay the tax on the orphanage. They're pursued by police officers, National Guard troops, Nazis and a country-western band, with much destruction along the way.
 * Crouching Tiger, Hidden Dragon, the two main female characters have one of these across the rooftops.
 * Total Recall. Benny drives Quaid and Melina to the Last Resort as Richter and his partner follow in their car, with both parties shooting at each other.
 * The beginning of Star Wars Episode IV: A New Hope. Princess Leia's starship is fighting with Darth Vader's Star Destroyer as she tries to escape so she can give the Death Star technical data to the Rebel Alliance.
 * Midway through Ong Bak, Tony Jaa partakes in one of these. Naturally, he is trying to avoid confrontation, but as soon as he gets cornered, he lays a serious beat down on any of those dumb enough to attack him.

Comic Books

 * Batman Under the Red Hood has several. The Red Hood always runs from Batman until their final encounter, usually fighting the whole way.

Professional Wrestling

 * ECW matches tended to go all over the arena.
 * WWE "Hardcore" matches often go backstage, and recently Kane chokeslammed Rey Mysterio Jr into the Gulf of Mexico after Mysterio Hurricanrana'd Jack Swaggar into the ocean. (The match started in an arena that's within spitting distance to the ocean.)

Video Games

 * This tends to be a recurring element in Sonic the Hedgehog games:
 * In Sonic Adventure 2, the final battle between Sonic and Shadow is a chase along an abandoned space station. The preceding levels are even called Final Chase and Final Rush.
 * Sonic Advance 2 was almost exclusively this style of Boss Battle.
 * In games with Super Sonic only final bosses, this tends to be the format they fall under.
 * The cutscene after final fight between Travis and Henry in No More Heroes features this, as a visual metaphor for them trying to "find that exit they call paradise"
 * In Legend of Spyro: A New Beginning, both fights against Cynder.
 * You'll probably have these in the Assassin's Creed series.
 * Mirrors Edge. AND HOW!
 * Then again, the REAL chase fight scene is with and you guys end up fighting on top a ship crate with Le Parkour action.
 * During an Escort Mission of a coach from Kariko Village to Hyrule Castle Town in Twilight Princess Link has to fight off several Bokoblins.

Web Animation

 * The climax of the second half of Bunnykill 5 involves

Western Animation

 * The climactic battle between Aang and Fire Lord Ozai in Avatar: The Last Airbender was this on both sides, at various times.
 * Any of the Family Guy fights that pit Peter against the giant chicken.
 * In the Popeye cartoon "A Dream Walking", Popeye and Bluto chase after a sleepwalking Olive while fighting over who gets to save her.
 * The final season of Star Wars Clone Wars features a long scene with three Jedi and Chancellor Palpatine fleeing from General Grievous and occasionally stopping to fight him.
 * In Thundercats 2011 Thundera's Gladiator Games in their thunderdome take this format: a race between two Catfolk competitors climbing, swinging, jumping and running up a giant tree to to ring a bell at its top, where violent kicking and punching in order to knock a competitor out of the tree are entirely acceptable methods of getting ahead. Young Prince Lion-O challenges Tygra to a match only to suffer a particularly humiliating Ring Out, landing in a pool of water below.