It can be done before or after a workout or at a completely different time.
As you noted some are very effective as part of a warm up/ movement prep.
The mwod is about improving your capacity to be in a good position during movement to improve performance and avoiding injury.
Some are best done following a workout.
Some will depend on what you are doing/ areas you are addressing.
Anything that hits deep would be done after working out or at a completely different time.

I was also wondering about this. Even if it’s done before a workout, should the body at least be warmed up to get the blood flowing first? I’m guessing doing the mobility work would be less effective if done cold.

Mobility work does not replace warm up.
You don’t need to worry as much about when you do it.
From the FAQ:
The MWOD is designed to be a discreet 10-15 minute daily practice in
addition to your normal workout routine. The MWOD can be done before or
after a workout or at night when you are laying on the floor watching
TV. Some of the mobilizations are very effective before workouts as part
of your warm-up and movement preparation strategies. On the other hand,
some of the mobilizations are best done post-workout or at a completely
separate time of the day. For example, the 10-minute squat test is not
appropriate as a pre-workout MWOD. However, rolling on the lacrosse
balls to mobilize your thoracic spine prior to an overhead workout is an
appropriate pre-workout MWOD. The bottom line – be less concerned about
when to do the MWOD and be more concerned about addressing your tissue
health and range of motion problems. Above all else, the MobilityWod is
about improving your capacity to be in a good position during movement
to improve performance and avoid injury.