Monster voluntary moves are processed first, at the beginning of each move, followed by Chip's voluntary moves, if any are made. Note that on the first move of a level in the MS ruleset, monsters will skip their turn.
An example: If there was a fireball about to step into the tile Chip is currently moving into, the fireball will move onto the tile first. When Chip's move is processed, he will step on top of the fireball and be killed.
Another example: In the images on the right, both Chip and the glider have already moved on the first turn. The glider than takes its action, and kills Chip before he can move.
The fact that monsters and Chip move on separate turns is what causes Chip to be unable to pass through a space between two monsters that are moving the other way through that tile.