Just to ensure I'm using the jargon correctly, I believe an 'Interface' is a groups of 'Pages' -- with that out of the way, I wonder if it would be relatively 'low hanging fruit' to offer the ability to move a Page to another Interface within the same Base? Why would you want to do this?
The Enterprise plan has granular permissions that control which Interfaces can be accessed by which users/groups. A Page in an interface might be useful to multiple groups, but that doesn't mean I want multiple groups to have access to the entire Interface. This results in having to re-create Pages in different Interfaces that have the appropriate permission settings.
A function to move a Page to a different Interface (within the same Base of course) would be a significant 'quality of life' improvement and time-saver in this situation. It could live in the page's '...' menu (see screenshot)
