Background:
The text says that at the start of the OL turn, all figures (starting with monsters, then heros) are moved 1 space towards the nearest pit space.

But it can sometimes make a big difference which figure is pulled before another figure, and which pit a figure is pulled towards, or even which space a figure is pulled into if more than one option is possible.

Questions and Answers:
Q1. Who decides which is the nearest pit space, if more than one have the same distance to the figure to be moved?
A1-1. The OL chooses which space a figure is 'pulled' to if more than 1 space is equally 'closer' to the pit. If there is more than 1 pit and some pits are equally the closest pit to the figure then the OL decides which pit the figure is 'pulled' too.
A1-2. The Heroes choose which space a figure is 'pulled' to if more than 1 space equally 'closer' to the pit. If there is more than 1 pit and some pits are equally the closest pit to the figure then the heroes decide which pit the figure is 'pulled' too.
A1-3. The controlling player chooses which space a figure is 'pulled' to if more than 1 space equally closer to the pit. If there is more than 1 pit and some pits are equally the closest pit to the figure then the controlling player decides which pit the figure is 'pulled' too.

Q2. Who decides the order of individual movements within the group of monsters and heroes?
A2-1. The OL decides the order in which figures are 'pulled', but all OL controlled figures must be pulled before any Hero controlled figures.
A2-2. The Heroes decide the order in which figures are 'pulled', but all OL controlled figures must be pulled before any Hero controlled figures.
A2-3. The Controlling players decide in which order their figures are to be pulled, but all OL controlled figures must be pulled before any Hero controlled figures.