Is there a limit on the charge that can flow in a wire?
But in AC, their movement is constrained. They move at 60Hz.


This shows a misconception. The AC frequency (60 Hz) tells you how often the current switches direction, not how fast the individual carriers are moving.


  If I hook up 24V, they move faster so more charge passes a area cross section for i=C/s. 


This is true to a point. But you have to remember that electrons are moving around randomly in all directions, and a reasonable current in an ordinary wire just "biases" the direction of this motion slightly in one direction or the other. If you have a higher current, the bias is even a bit more in the direction of that current, but not every electron is moving with the current and the effect on the speed of each electron is fairly small.


  I assume when I hook up a 12V battery to a wire, all the free electrons in the wire move.


This is not really true. All of the electrons were already moving. Applying a voltage just made the average of that motion slightly to one direction or the other.


  Is there a limit on the charge that can flow in a wire?


Practically, the maximum current in a wire is limited by the resistive self-heating of the wire. If too much current is carried, the wire will get hot enough to melt itself or cause a fire hazard. Typically we choose a wire large enough to limit its temperature rise to 20 or 30 degrees C under expected loads. On this basis, you can find tables online giving the ampacity of round wires depending on diameter or pcb traces depending on width and copper thickness.