Why does the sound in my apple cinema display stop working?

First your displays speakers (f you have the newer Cinema display) are feed from the displayPort connection via the Thunderbolt port off your system. So when you have a USB connection in addition to the displayPort's you've confused your computer which path to send the sound through!

So what happens here is your sound is taking a different route to the speakers which is slower! Which is why the sync is off between the mouth movement and the voice.

You should only have the displayPort connection of the display which has speakers. If you are using external speakers then you'll want to then use the USB port off of the display not the system to drive them. Make sure you've directed the sound output correctly by checking your Pref settings. If your second Cinema display also has speakers make sure they are disabled.

The last issue with out of sync sound can be the codec being used, make sure you have the latest version. In some cases your system may have resource issues. Make sure you have at least 1/4 of free space on your boot drive (internal SSD drive). I would also monitor your systems RAM usage. I like this app: Memory Clean.