Cuccos appear as items. If the Cucco is grabbed and thrown at a player, it will call its flock and start attacking that player relentlessly, although other players can be hurt if too close to them. If a Cucco is not picked up and hit by a player, they will start attacking the player that last hit it instead.
Now there's a creature that's just poultry in motion. It usually minds its own business, but if you attack it, the chickens will come home to roost--by body-slamming you over and over...and over... A pretty fowl tactic if you ask us. It's almost futile to fight back, and eventually they'll lose interest and stop driving you Cucco.
Trophy Description (Wii U)
Why exactly a Cucco flees is anyone's guess. Maybe it knows what will happen when it is attacked and is trying to save you from that boundless rage. Once angered, it will call its flock and attack the last player to hit it. If it's thrown and hits a fighter, whomever it hits will suffer the wrath of the flock.
SNES - The Legend of Zelda: A Link to the Past (04/1992)
N64 - The Legend of Zelda: Ocarina of Time (11/1998)
Even though Cuccos are 3D when appearing as an item, in Smash Run their models are pixels instead. The same goes with the Lakitu from the Assist Trophy (which is a pixel model) and a Lakitu in Smash Run, which is in 3D.