Working on a site that had #2 comment's patch applied, we faced a problem with the automatic download after the batch finished. More specifically:
The files were properly saved in the private directory, however the respective database entry in file_managed table was not updated and pointed to the temporary directory.
Thus $headers in finishBatch() were empty because the function failed to find the file via its URI and consequently it was not downloaded.
Based on that patch, I added some code in order to update the respective database entry with the new URI (that points to private directory), along with a couple of log messages to have a better overview of the process.
