Family Rivalry

Two rivals' children compete against each other in the same competition the rivals had competed in. One of the kids is a major character, whose parent lost in that competition (and this fact will be told for exposition purposes at the start of the story). Story usually ends with the major character avenging the senior's defeat, though the subversion of a repeat loss is also common, especially in comedy series. Or, a Dark Horse Victory may come along, and both sets of kids lose.

See also Feuding Families, which is a much more serious, often bloody, and frequently tragic version of a family rivalry.

Anime and Manga

 * Dragon Ball: A few generations removed, but this is what happens to Goku's great, great, great grandson and Vegeta's Great, great, great grandson, who meet for the first time in a martial arts tournament and both have inherited their ancestors love of battle.
 * Initial D has a race between Fujiwara Takumi and Kogashiwa Kai, whose fathers (Fujiwara Bunta and Kogashiwa Ken) had raced each other years ago.

Western Animation

 * Hey Arnold!!, "Tour De Pond", with a boat race
 * Rocket Power, "Race Across New Zealand", with a sports competition in New Zealand
 * Step by Step: an ep of this show has one of the daughters trying out for head cheerleader. Towards the end, the mother tells the daughter about how she was robbed when she tried out for that spot... and as it turns out, the spot goes to the the daughter of the woman who won the spot when mother tried out. //character names welcome, please