ETC will die after ETH PoS i think. Such hash from ETH will destroy any coin.
And ASICs will be reconfigure for ETC, i think it is rather easy. But i dont think that the PoS will be the end of the GPU mining. We`ll see new coin, better than all others and forgot ETH.
It won't die if everyone switches from ETH mining to ETC mining, the daily supply is going to be still the same. This is called difficulty. Basically the more people mine it the reward needs to be split equally against everybody else.
Also I am pretty sure most ETH ASICs won't work with ETC, I know that the E3 can be reflashed to work but some of the other ASICs won't work with ETC right off the bat. The developers made a small change to the algo to get rid of ASICs.