Assist characters come from Assist Trophies that appear on the stage, considering they are turned on. There are four types of assists: Offensive, which use they're powers to attack opponents, defensive, which will protect the player, item, which spawn items, and accidental, which may hurt the opponents as well as the one who summoned him/her. There are also assists that will do random events each time.
Once summoned, Kamek will fly on his broom to the top of the stage and will do one of three things: Cast a fire spell on opponents, conjur up a helpful item for the summoner, or create an earthquake that hurts everyone, even the summoner!
Eggman is a semi-rare assist. He appears in his Egg Car. He will float around the stage, firing missiles like crazy! The knockback the missiles cause to the opponents depends on the size of the missile.
Leon and Panther appear back to back, and then they hop off and land into they're Arwings. Once in they're Arwings, they begin to rapidly shoot the stage from the background. Leon's shots are quick but deal less damage, and Pather's shots are slow but deal more damage. They are a rare assist.
Ridley is one of the largest assist trophies in the game. Once summoned, he will roar, and then fly to the background of the stage, spitting out large fireballs, which deal burning damage to opponents. He is a rare assist.
Shadow will use his Chaos Control to slow down the opponents, making it easier to attack them. He is a semi-rare assist.
Pokeballs are just like assist trophies, only smaller, rounder, and release Pokemon instead of characters. Just like assist trophies, the Pokemon can be offensive, defensive, item, or accidental. Some are locked, and are unlocked in a random order.