I've done a bit more research - I'm not a developer so this is new to me - but it seems like the functionality is called the iOS "Document Picker".
Info here: https://developer.apple.com/library/con ... ction.html
One of the limitations of the current implementaiton is that only a single file can be selected and moved/copied. I believe the "Files App" in iOS 11 enhances this to allow multiple files to be selected, and I think it's an extension of this same API.
That is, if Luma Fusion is extended to use the Document Picker for import and export, all this will come for free with the Files App.
It's hard to find Apps in the App Store which highlight their use of this extension for file management, but it's the usual suspects such as DropBox, Google Drive, OneDrive for their own cloud storage. For NAS storage, I've found DS File, DS Cloud for Synology, and FileBrowser for Business (any SMB or AFP share on the network).
Hope that helps. I can't believe it's been there all along and I've never found it.