Thanks for your question. I regret that the current MLA version does not have any features that support an export/import project of the type you outlined.
Media Library items, their assigned taxonomy terms and custom field values are identified by integer IDs automatically assigned by the database manager. These IDs are only unique within one database, so they can’t be used as-is for moving items to another database. If the taxonomies and terms already exist in the target database, it’s hard to decide how to map terms coming from the source database to those in the target database. These are two of the technical considerations that make a good solution complicated.
The topic has come up before:
Export / Import Att. Categories
export / import settings
What about a media library export/import function?
Someday I may find the time to dig into this area but I can’t promise anything soon. In the interim, here are some other plugins to investigate:
WP All Export and WP All Import
WP Attachment Export (see the note about WordPress 4.4+)
FrédéricGilles.net – Migration tools for WordPress
Frédéric Gilles has a number of migration tools – perhaps he has thought about the WP-to-WP case.
Have you tried the built-in WordPress 4.4+ Export and Import tools? If so, what are they missing? Maybe there is an enhancement I can make to complete the solution.
Thanks for your question and for your understanding of my answer. I regret that the news is not better, but I am marking this topic resolved simply because your question has been answered. I welcome any further comments you have on the topic. Thanks for your interest in the plugin.