When using Organize Boxes, there are three buttons at the top that determine how to move Pokémon, the red (o), blue (o)<->(o) and green [88]-> buttons.
Obviously, the red button is for normal movement and the green one allows you to select multiple Pokémon at the same time. However, I can't for the life of me figure out what the blue button is supposed to do. It seems to do the exact same thing as the red button.
What am I missing?
Best Answer
After more experimenting, I managed to stumble over the answer. For future reference, I'll just post it here.
The regular red mode allows you to tap a Pokémon to bring up the menu (Move, Summary, Item etc.) while blue does not; it's used exclusively for moving. More significantly, when using button controls, selecting a Pokémon in blue mode skips the menu and immediately prepares the Pokémon for moving.