Moving blogs.dir location on existing multisite

I’m sure I’ve missed a codex article somewhere, but on an existing multisite install, our media is in /wp-content/blogs.dir/, as usual. We want to move this folder to a different location, e.g. /media/.
After moving the folder, what are the steps required to tell WordPress (and all of the existing sites) of the new location? So far, I have set:

define( ‘UPLOADBLOGSDIR’, ‘media’ );

This doesn’t work on itself. New files are uploaded to the media folder, but when looking in the Media Library, all of the images are broken.
Per ms-default-constants.php, it looks like BLOGUPLOADDIR is never set. Is this related? Where and how would I re-define BLOGUPLOADDIR; sunrise.php?

Then after all of this, I assume I have to go into each sites’ options table and change the upload_path key manually? (or via Network Admin -> Sites > Edit, etc)