In the chase rules, the chaser and the chasee make piloting rolls and you compare the results to see whether the distance is opened or closed. My question is when do you make these rolls? Do they require an action during the initiative-ordered turn from each pilot? If so, when does the distance change? If they happen outside of the pilots' turns, when do they take place? Do they still consume an action? Thanks in advance!

The check takes place at the beginning of each round before anyone takes their turn, and that's when the ranges change.

They do not count as the character's action. So you can use the chase rules to do a running gun battle or some other highly mobile combat scenario.

Awesome, thank you!