Any system is eventually relaxing to equilibrium, which is understood as the set of its most probable configurations. For a system interacting with the environment, this usually mean settling to the state with the lowest energy. Thus, electrons in the metal keep moving until they reduce their potential energy, that is until they neutralize the electric field inside the metal.
How exactly this relaxation happen, and how to describe the resulting equilibrium state is the subject of statistical physics (or of thermodynamics, if we accept the view that any system relaxes to equilibrium as a law of the universe, without going into details.)
Update:
1.
The following deductions are made assuming equilibrium is always reached in any shaped conductor
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My question: is why wouldn't electron not reach equilibrium and continue to move in search for equilibrium? By this, I mean, why would they necessarily experience net zero force after some time?
Note that we are talking here about an isolated conductor. E.g., if we were talking about a wire connected between the terminals of a battery, the charges in the wire would not reach equilibrium, but would continue to flow through the wire from one terminal to the other. Eventually, the battery would discharge and the current stops - e.g., we do reach equilibrium, but the equilibrium of a bigger system (wire + battery + other possible elements in the circuit.)
2.
I mentioned previously thermodynamic laws. One of them is the law of increase of entropy, which essentially states that the system always moves towards a state of thermodynamic equilibrium. From the point of view of statistical physics (i.e., from the microscopic point of view) it means that the system eventually explores all possible microscopic configurations, and can be found with equal probability in any of them. If we started from a specific configuration (or a set of configurations), then this configuration becomes less probable with time, and for realistic systems this probability becomes vanishingly small.
Finite resistance (i.e., friction) in a conductor means that moving charges transfer their energy to the lattice. The energy thus becomes redistributed between that of the electrons (potential and kinetic) and that of other degrees of freedom (lattice, energy of the random motion of electrons and nuclei, etc.) Having a state in which electrons have any remaining potential energy thus becomes very improbable.
A toy model to convince oneself that en electron in a potential eventually comes to an equilibrium is Newton's equation for a particle in a potential with friction:
$$
m\ddot{x}=-\gamma \dot{x} -\frac{dU(x)}{dx},
$$
where viscous friction term $-\gamma \dot{x}$ models the electric resistance. Solving exactly for a simple potential, e.g., $U(x)=kx^2$ shows that electrons eventually stops at the minimum of this potential. On the other hand, potential $U(x)=Ex$ does not have minimum - it could correspond to a wire connected to the battery, maintaining constant field gradient across the wire.
(Of course, in this simple model I neglect the potential created by electrons themselves.)
