Catch That Pigeon

Animated plots frequently involve natural enemies chasing or hunting each other.

Named for the Animated Series Dastardly and Muttley in Their Flying Machines. In that case, it's enemy pilots and an Instant Messenger Pigeon.

Can be Road Runner vs. Coyote if we see them comically failing repeatedly to catch it.

Eastern Animation

 * Wolf and Hare from Nu Pogodi.

Magazines

 * Pwt and Crraw in Muse Magazine.

Video Games

 * Sid & Al's Incredible Toons is a video game example.

Western Animation

 * Tom and Jerry: The cat, Tom, is always chasing or otherwise trying to thwart the mouse, Jerry. It rarely ends well for Tom.
 * Looney Tunes and Merrie Melodies:
 * Sylvester and Tweety. The former, a cat, really wants to eat the latter, a canary, but never succeeds.
 * Wile E Coyote and The Road Runner. Wile E. Coyote tries repeatedly and fails repeatedly to catch the Road Runner so he can feast on him.
 * Affable doofus hunter Elmer Fudd and Bugs Bunny and/or Daffy Duck.
 * As mentioned in the description, the Trope Namer is Dastardly and Muttley in Their Flying Machines.