There should be a easy way to get a list of pages moved out of a category within a certain period.

use case: empty category deletion.

On most wikis, policy requires admins to check for 'category de-population vandalism' before deleting an empty category. e.g. https://commons.wikimedia.org/wiki/Commons:Deletion_policy#Categories

On some wikis, an admin needs to determine whether the category has been empty for a policy defined period of time before it can be deleted.

e.g. on English Wikipedia, it is four days. https://en.wikipedia.org/wiki/WP:C1

That policy can only truly be complied with if the admin has a snapshot of the entire database four days prior to the deletion.

As far as I know, the best way to approximate this is to look for likely members of the category, and check their history, and check the history of the editor who tagged the category for deletion.