Each sentence describes a battle between two things which differ by a single letter. The pairings are as follows:
A SHOUT defeats a SCOUT.
A CLOAK breaks a CLOCK.
ROT overtakes ROO.
A FORK routs YORK.
CURIE breaks a CURSE.
A CHECK overwhelms a CHICK.
A BOWLER takes out BOWSER.
A HERD tramples the HERO.
A PASTRY stymies a PANTRY.
A FILM overwhelms a FILE.
The changed letters in the winners spell out HATFIELDSM and the corresponding letters from the losers spell CCOYSISONE. Taken together, these spell out the clue phrase Hatfields-McCoys is one. The Hatfields and McCoys gave their name to an infamous FEUD.