Voltage will affect as it is the 'strenght' of electrons to be moved, but it is not the question (I don't explained myself enough), who is, if the exagerated difference in the rate of a capacitor (the isolation capacity), for instance, from 50 to 500, could affect the circuit behavior? That is, if the much bigger capacity of isolating could do that a capacitor with a 500 V rating, feeded at a regulated just 5 volts (instead of the maybe 220 V for wich it could have thought) could give us for instance a real 1.9 nF, instead of a 2.2 nF marked in it's specification, making a different oscilating time for the circuit than the designed one, or, on the other hand, it will remain offering the same capacitance of its specs. independently of the so different isolating rate.
- - - Updated - - -
The associated question could be that if there are two type of capacitors, for instance, electrolitic and tantalum, both specified as 1uF, or maybe tantalum and ceramic at maybe 1nF both, what difference could it do in an given oscilator?