Sakurai details a couple of the battle stages players will see throughout this mode in the screenshots below, including Run! and Climb! areas that give certain advantages to different characters depending on their default speed or jump height and length. Here’s what the director said on Miiverse:
Here’s one of the final battles in Smash Run, called “Run!” As you’d expect, it involves you dashing toward the finish line. In general, the fighters who are fast by default, equipped with speed-boosting equipment, and powered-up with speed boosts over the course of that Smash Run will have the advantage. However, speed isn’t the only factor that will determine who will be victorious…
Those red things in the picture are called danger zones, and if you touch one, you’ll get penalized by respawning farther back in the stage!! The danger zones disappear after someone else touches them, so fighters following those in the lead will have an advantage. You can still find speed boosts throughout the stage, so it’ll be a heated battle ’till the end!!
Here’s another final battle called “Climb!” Obviously, those that can jump or fly higher will have the advantage, but they also have to avoid the danger zones!! It gets pretty challenging if you’ve boosted your speed too much, as well. Everyone really gets into this when playing multiplayer.