I've got a 0.9.4 downloaded on a computer I no longer want to use. Can I send those coins to the 0.10.0 chain?
The chains are exactly the same. V0.10 continues on where v0.9 left off. Actually, right now you can use either version of the software, com January you will need to use v0.10. This is not because v0.10 is on another chain, but rather because in January v0.10 will start enforcing some rules that v0.9 won't understand.
The way your address and wallet is built is all backwards and forwards compatible though. The wallet itself might quit functioning some day, but the keys are still good. You just need to upgrade the software.
The chains are exactly the same to begin with. You can use either 0.9.x or 0.10.x until January 2017 with no change in usability (although 0.10.x brings some improvements). After January, the keys files will be the same, but you will need the new version of the software to interpret the new chain properly.