Do MacBooks slow down when not connected to power?
Yes. Intel calls it "SpeedStep." Variable CPU clock speeds that adjust to conserve power based on demand. The feature is engaged whilst the device is on battery power. All Apple products support this feature (it's actually supported at the hardware level and the operating system's kernel) but since desktop devices don't have batteries, you'll never experience it on anything that isn't mobile.

Your MacBook Air supports SpeedStep, as does the iPhone, iPad, and the traditional MacBook Pro.

More info on SpeedStep: http://en.wikipedia.org/wiki/SpeedStep#Mac