Thoughts
What would it look like if you followed the moon?
If you followed the sun, you’d move west close to the equator, and it would always be noon, and the sun would always be directly overhead.
If you followed the moon, it wouldn’t always be night, though. You’d get to see the phases of the moon dynamically change, instead of waiting until the next night. You would probably veer far from the equator to keep the moon over head.