So you dutifully back up your old iPad (or iPhone), connect your sparkling new iPad Air, and do a restore using iTunes. You fire up Spotify and all of your local files don’t play! WTF? They look ok, but you click and click and get nothing.
The fix, albeit lame, is to delete the app and reinstall it while connected (via WiFi) to your desktop Spotify installation. You’ll also have to enable the “Available Offline” option on the applicable playlists.
Not the best solution, but it works.
2016-04-04: Another way I’ve had success with is on your computer version of Spotify, make a playlist of your local files. (The easiest way to do this is press Ctrl+A to select all songs, right-click and choose Add to Playlist.) Bring that playlist up on your mobile device and enable the Available Offline switch.