Steps to reproduce the issue
The new category should have the changed permission in it too.
It doesn't. It keeps the default.
The same applies to batch copy for articles too.
The batch move keeps the permissions.
There is an option during batch process that refers to access levels but that's the viewing levels stuff. So as long as there is no specific option shouldn't copy a category mean we copy the specific permissions too?
Indeed. Code does not check/copy for the specific rules applied to the original item(s) (in all batches, not only categories or articles but also menu items, etc.) It could be done but implies quite some work.
No Code Attached Yet
It should keep as many values as possible. If I remember correctly, it's also an issue with nested categories and menu items losing parent item relationships, but it's been awhile since I've had to do that. It would be very helpful and help users avoid more steps.
I can confirm in Joomla! 3.7.0-RC2 for Category and Articles.
For me we need a message in the batch procedure that explains the copy not keep acl. While now appears "Set Access Level --> Keep original access levels" which can be misleading.