In a metallic conductor, electric current is thought to be due to the movement of:

The movement of the charged particles is what causes the current in any metallic conductor. Ions or any other charge particle that is freely accessible for travel through the conductor can cause current to flow.

As we are aware, the current formula explains that it results from a charge moving through a conductor once every unit of time. 

This is,
$I=\frac{Q}{t}$, where I is current, $Q$ is the charge and $t$ is the time.
It is evident from the description above that the movement of the charge is what causes current to flow in a conductor.
In metals, protons are bonded, leaving an electron as the only free charge particle. As a result, one electron exists as a charge particle that may influence how current moves through metallic conductors.
So, it is believed that the mobility of electrons in a metallic conductor is what causes electric current.

Hence, the correct answer is option C.