In API v2, you can look up the information for a "deleted" file (including a file which was actually just moved) using /2/files/get_metadata:
https://www.dropbox.com/developers/documentation/http/documentation#files-get_metadataIf you set include_deleted=true, it will return the metadata for the file at the old location. Note that this will give you a DeletedMetadata object, not the full Metadata object.
If you store the file's ID though, you can get the full current metadata, regardless of its current path, by supplying the ID in the 'path' parameter to /2/files/get_metadata.