According to the move rules, you can make a normal move as long as you are on the battlefield floor or on a platform and you must remain on the floor or platform unless you do a jump/climb/fall/etc. You can't stand on anything that is less than 1"x1", but you can move over it as a climb (says you can climb horizontally across obstacles, climbing only ends when center point is on the floor or a platform).
So you can move along a wall top (as a climb), even jumping between them, but you can't end your move on top of them without falling.
Your model is considered on the platform as long as its center point is on the platform. This allows models with large bases to use platforms smaller than their base. This is explained in one of the illustrations in the movement section. If you end a turn with the center off the platform, you fall.
It's worth taking a moment at the beginning of the game to define the terrain features, like the platforms, obstacles, doors, stairs, deadly terrain, and so on.
Objectives are only determined at the end of the game round. See the box on pg 37. One thing that wasn't immediately obvious is that once you have claimed an objective, it remains yours until your opponent claims it.
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