I don't know about metals, but in insulating dielectrics there is no bulk current (as you seem to suggest by the "mobile charges". ) like there is in a metal. Instead, the imposed electric field distorts the electron clouds in the crystal structure of the dielectric, the electron densities moving toward the dielectric's positively charged face. As a result, the opposite face of the dielectric is electron-deficient. Hence, an electric field in the adjacent space is induced is induced by the partial positive charges on the dielectric's face. In electric capacitors, there is a metal electrode on the electron-deficient side as well. When the external EMF at this face is negative, the charge displacements in the dielectric and the imposed field form a stable system that retains charges in the electrodes and charge displacements in the dielectric for some time - theoretically forever. At the risk of repeating myself: In the ideal case, there is no current flowing through the dielectric when a capacitor is charged and discharged. Any observed current is strictly virtual. Charges flow into and out of the device's conducting plates without crossing the dielectric. If you want, calculate the instantaneous currents in a resistor and a capacitor connected in parallel. The capacitor's apparent current equals the product of its capacitance and the time derivative of the imposed voltage. The current through the resistor is given by ohms law, I= V/R. The relative currents (say, at time t=0) will depend on the frequency of the voltage source and the capacitance versus the resistance. Assume ideality: There is no resistance in the capacitor and no capacitance in the resistor, and no inductance anywhere. Over some ranges of these variables, the capacitor will present a virtual short circuit across the resistor, especially at t=0 sec, when the capacitor is charging at its fastest rate.
If you're still interested in electronics applications:
This effect is evident when filter capacitors in DC power supplies retain electric charges for some time after the power is turned off, which is why some electronic devices take as much as a second or so to turn off after their power is cut off. The effect is also responsible for the hazard presented by high-voltage charges on filter capacitors after power is turned off. In some cases, when large capacitances are involved, a dangerous charge can be retained by the capacitors for hours. Since real capacitors can 'leak' charge across their electrodes, the devices eventually lose their charge.
Another contribution to the dielectric constant is provided by changes in the dielectric's atomic structure itself - changes in the position of atoms and direction of chemical bonds. Since the imposed electric field is moving atoms that are thousands of times more massive than electrons around and between the atoms of dielectric, this contribution to the dielectric effect lags behind the distortion of the dielectric's electron clouds. The bulk manifestation is often referred to as a 'memory' effect. Such memory effects would be desirable in capacitors, like 'supercapacitors' with capacitances in the Farad range, but when timing of a capacitor's response to a changing electric potential must be accurate, the memory effect can result in failure in circuit design. There is a specification for the memory effect that manufacturers of capacitors list in their data sheets. Along with resistance to leakage, breakdown voltage and other specs, the magnitude of a dielectric's memory effect is the reason designers will specify the type of dielectric (mica, teflon, polystyrene, ceramic, polyester, polypropylene and less common materials) to be used for some capacitors in their designs - i.e. when timing is crucial and the current frequencies are high enough, capacitors with a small memory effect are specified. "Practical Electronics for Inventors" by Scherz and Monk is the reference I turn to for explanations of the various non-ideal properties of capacitors.