The Chase

A plot centered around characters pursuing one another.

A drawn out chase which forms an underlying theme for a series is a Stern Chase.

They Have the Scent is common in short chases.

Examples of The Chase include:

Film

Honored Enemy of The Riftwar Cycle in which soldiers on the opposing sides of a war band together to escape the dark elves chasing after them to kill them for a variety of reasons.
In Robert E. Howard's Conan the Barbarian story "Black Colossus", after his defeat, Natohk abducts Yasmela. Conan chases after them.
- In "The Gods of the North", an obviously magical woman lures Conan into a chase that brings him to her giant brothers. When he kills them, he chases her again—this time, she is seriously frightened.
In Poul Anderson's A Midsummer Tempest, chasing the escaped Rupert is half the book.
The Time Scout series is full of these, starting with the second book. Mostly they involve Skeeter Jackson.

The William Hartnell Doctor Who episode "The Chase," in which the Daleks first develop time travel and chase him and his companions all through time and space.

Dick Tracy had this as a major element in his stories as the police detective relentlessly chases a particular villain who gets more and more desperate as he realizes to his terror that this cop will not stop.

Snuggleninja is always chasing the butterfly.
In Sinfest, after Death realizes the zombie's a fugitive, the chase is on.
- Also, annually, Death has to chase down the year, who always tries to make a break for it.

Pepe Le Pew and his kitty.
Roadrunner & Wile E. Coyote.
Bugs Bunny and Elmer Fudd/ Yosemite Sam/ various unnamed characters, Sylvester and Tweety Bird/ Speedy Gonzales/ various unnamed characters, Tom and Jerry, Dick Dastardly and Yankee Doodle Pigeon, Gargamel and the Smurfs, etc., etc., etc., etc., we get it, cartoons involve chasing.
The entire plot of the Danny Phantom episode "Reality Trip". The Guys in White constantly pursue Danny and friends after Danny accidentally exposed his half ghost power to the public.
This troper can't recollect a full episode devoted to this, but is willing to bet that The Simpsons have done it.
The episode "The Chase" of Avatar: The Last Airbender stands out among numerous episodes dealing with the Fire Nation's pursuit of the heroes.