Chests       : Never
Locks        : Never
Other Items  : Pushed
Slime Blobs  : Never
Sharks       : Never
Dragons      : Never
Runners      : Pushed/Willing
All Guns     : Never
Other Foes   : Pushed
Robots       : Willing (Pushed if pushable)
Mines        : Never
Sensors      : Pushed/Willing
Missiles     : Never
Seekers      : Pushed
Bullets      : Never
Spitfires    : Never
Scrolls      : Pushed
MovingWalls  : Willing
Boulders     : Pushed
Boxes        : Pushed
Crates       : Pushed
CustomBoxes  : Pushed
CustomPushes : Pushed
Pushers      : Never

NOTES:

-Whatever can be pushed can be pushed through a transporter as well.
~"Willing" means they went through the transporter through natural behavior (most things) or were directed through Robotic to go through (only applies to Robots and sensors).
~Nonhostile fish and invincible ghosts can only be pushed indirectly, not directly by the player. Current cannot force fish through transporters.
~Sliders cannot traverse lava or goop.