Suppose a charged capacitor (parallel plates), the negative and positive charges on two plates attract each other. Which force cause the negative charge carriers (electrons) move through the circuit to the other plate if we connect two plates (from outer sides) with a wire?
Every system likes to decrease its electrostatic energy.
The charges on the plates are almost in stable equilibrium. The charges on the opposite plates attract them, and the charges on the same plate repel them with almost the same force.
However, a capacitor has fringe fields:
These may be negligible when calculating the field inside a capacitor, but they are extremely important when there are wires in play -- by $\vec J=\sigma\vec E$, for a wire (which has high $\sigma$), even a small $\vec E$ can create a large current.
And these create the tiny perturbation required to push the charges out of their almost-stable equilibrium.
Note that redistribution of surface charges occurs on the wires as well, so it's not just the fringe fields pushing the electrons.