If a perfect conductor were to move, what happens to the electrons? Theoretically, if a perfect conductor were to move (or start spinning), would the electrons inside it remain where they were or be 'dragged' along with the conductor? 
 A: The electrons are charged particles and will experience a Lorentz force if the conductor is moving orthogonal to a magnetic field, causing one part of the conductor to be more charged than the other. If the conductor is uncharged and spinning, it will exhibit the Barnett effect, but this has more to do with other properties of the body.
You may be thinking of a London moment, where a spinning superconductor generates a magnetic field. Normally, electrons are kinda randomly going all over the place in a conductor, but in a superconductor, electrons can form Cooper pairs, causing them to be attracted enough to stick around in roughly the same spot with regards to the body. The electrons in the body create current loops and a magnetic field directed along the spin axis in accordance with the Biot-Savart law.
A: ...Just in case you are asking a more basic question than the other answers have assumed:
You might be operating under a common misconception about how electricity flows in a conductor by thinking about it like water in a pipe.  Obviously, if you have an open-ended pipe with water in it, and you spin the pipe around, water goes flowing out.
Electricity doesn't work that way, and the electrons don't actually "flow" through the conductor. What flows through the conductor is electric charge. The electrons do move, but only barely, and that movement is called drift:
http://en.wikipedia.org/wiki/Drift_velocity
...meanwhile the electrical charge itself is transferred as a wave at speeds that approach the speed of light.
Drift is proportional to voltage - in your question I believe you are picturing a conductor that is not connected to anything. Swinging it around will not cause the electrons to come spilling out - they will be dragged along, as you say, since they are part of the conductor itself.
In fact, the better the conductor the less the electrons should move (since perfect conductors have no voltage loss thus no drift).
Bullet point fun facts:


*

*The electronics will actually be travelling backwards to the flow of current (IIRC - only 50% sure of this).

*If the current is alternating, then the electrons will have no net movement along the conductor - they'll go back and force around where they started.

