I have several Xbox 360s and I use one hard drive to move my content around. Once Bring Down the Sky is installed from the bonus disc (the 2-disc version, Platinum Hits), will I be able to  play it on a different machine then what it was installed on?

If it has been installed on one hard drive, and you use that same hard drive to play on other 360s I would expect there to be no problems. 

Content is not tied to the machine itself, just the hard drive, so when you install data or content from a bonus disc in your case, the data is only written to the hard drive.  For Marketplace content, that is tied to your gamertag or LIVE account so you won't have to keep repurchasing it.

However with that said, don't quote me on any of it because I could always be wrong.
If someone else has more insight, have at it lol.

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In the end we're all just chalk lines on the concrete
Drawn only to be washed away