From Bulbapedia, the community-driven Pokémon encyclopedia.
Cobalion surrounded by Rusty Pokémon
Rusty Pokémon (Japanese: サビたポケモン Rusted Pokémon) are Toy Pokémon in Pokémon Rumble Blast and Pokémon Rumble World that are rusted. Rusty Pokémon have the Rusty Special Trait. When Rusty Pokémon are turned into toys and recruited, they will only know the move Struggle, and cannot learn any more until they are cleansed. The Rusty Special Trait does not come in Rusty+ or Rusty++ versions.
In Pokémon Rumble Blast
Rusty Pokémon can be cleansed by going to any one of the Glowdrop fountains in Toy Town, Easterly Town, Westerly Town, Northerly Town or Axle Town. Once they are cleansed, their moves will be available and will have a Trait+.
When releasing Pokémon to obtain new evolutions in Pokémon Rumble Blast, seven of the same Pokémon need to be released to obtain the respective evolution, but Rusty Pokémon do not count towards the seven unless they are cleansed.
Additionally, when a Toy Pokémon comes into contact with Dark Rust, it will rust as well; however, instead of simply being limited to Struggle, it will be covered by a purplish-black aura and be controlled by Dark Rust, becoming a part of the Dark Legion. Pokémon infected by this kind of rust are always hostile, and will not drop as toys. Any Toy Pokémon rusted this way are not able to speak verbally.
In Pokémon Rumble World
Rusty Pokémon start appearing after the player reaches at least Adventurer Rank 55. By clearing the "Dragon Marshal Advance" Challenge, the Rusty Removal Spot will open in town, which will cleanse Rusty Pokémon. The first wash is free, but each subsequent cleaning requires 1 Poké Diamond per Rusty Pokémon. Multiple Rusty Pokémon can be cleansed at the same time. Any Rusty Pokémon in use cannot be cleansed until switched out. Additionally, if a Pokémon is selected as a Favorite or Partner Pokémon, then it cannot be cleansed until this status is disabled.
Rusty Pokémon are indicated with a brown name, as opposed to pink for other singular Special Traits, or blue/purple for multi Special Traits. Once cleansed, they acquire a multi Special Trait, and like in Blast, their moves will be available.