I just tested out this behavior in our SP 2013 on-prem farm and wanted to share it in case someone else is seeing the same:
For publishing pages, as long as the page is checked-out or checked-in as draft, the drag & drop functionality works, however as soon as you publish the page, it breaks. I was able to confirm this with multiple libraries and multiple page types.
The only workaround I found was to use a web part page which doesn't use the versioning infrastructure and on this one drag & drop does work all the time.
Is it my environment or a known bug?