I had the first 2 episodes of TWD and they worked perfectly, I completed them then eagerly awaited the release of ep3 and promptly downloaded it when it came out.
My first problem was that the game kept freezing during the opening cutscene of episode 3. I tried replaying through episode 2 and letting it follow on from there but it still froze at the opening cutscene.
I tried re-downloading episode 3 and had the same issue, followed by re-downloading ALL episodes to no effect.
I deleted the entire game, cleared the cache on my hard drive, the re-downloaded ALL episodes for a clean install and was prompted to download an update when the game started up.
When I accept the update the system then told me that the device containing the game has been removed, when I decline the update I'm signed out of Xbox Live and returned to the dashboard.
So I don't get some of the same questions I've already encountered here's some more info.
I downloaded ALL episodes on the same console, using the same profile/gamertag. I have not transferred any information between consoles.