Use the following moves when playing as the indicated character to knock your opponent off the platforms.

Mario and Luigi:
Press B to throw a fireball when they try to come back.

Ness:
Press B for the PK Fire if they come near your character in the air. Your opponent will be pulled down and paralyzed once it hits.

Yoshi:
If you are near the end of the platform, wait for your opponent to approach. Then, press B to turn them into an egg. If you are close enough to the edge, the egg will make them fall and they will not be able to get back up.

Donkey Kong:
After knocking an opponent off, press B to charge up the punch. Use the punch when they are close enough while in the air.

Kirby:
Press Up + B near the edge when your opponent approaches.

Link:
Press B to throw the boomerang after knocking them off the platform and they try to get close again.

Fox:
Press B to keep firing the blaster.

Samus:
Press B to charge up the gun, then shoot it when a knocked off opponent returns.

Pikachu:
Jump and press B to shoot a little ball of electricity. If it hits a returning opponent, they will be paralyzed.

Captain Falcon:
Press B to use the Falcon Punch when your opponent tries to return.

Jigglypuff:
Press B to use the lunging punch when a returning opponent is close enough.