Can you explain me what would happen inside the conductor, if the conductor is not in electrostatic equilibrium?
If electric field inside a conductor is not zero, then the conductor is not in electrostatic equilibrium.
Let us take a conductor and apply a potential difference V across it as shown below.
Now the electric field inside the conductor is not zero. Hence, there is no static equilibrium inside the conductor. Since , free charges inside the conductor start moving in a direction opposite to the electric field, thus producing conduction current.