Some reading material
High-K ceramic dielectric materials (X7R and the like, not C0G) are ferroelectric: they contain dipoles which can align with the electric field, which increases permittivity, and thus capacitance.
At zero voltage, zero electric field, and zero stored charge, all these dipoles are in random orientations*.
As the electric field increases, more dipoles will align with it. This means there are less "free" dipoles in random orientations remaining that could align with the electric field, thus permittivity decreases. When the field is strong enough that all dipoles are aligned with it, the material is saturated and behaves like a non-ferroelectric dielectric, it no longer helps increasing the capacitance.
These materials are great because they have very high permittivity, which means a lot more capacitance in the same volume than a non-ferroelectric material like C0G/NP0 ceramic, film, etc.
However permittivity (and thus capacitance) will decrease as the electric field gets stronger.
At the same voltage, a physically smaller cap will have a larger electric field between its electrodes, since electric field (in V/m) is voltage divided by distance between the two capacitor plates.
Therefore the smaller capacitor (0805) loses capacitance faster as voltage increases than the bigger cap (1210).
It's not just about the footprint size (1206 vs 0805) but more about dielectric volume, as the total number of dipoles available in the material is proportional to volume. So a capacitor with the same footprint, but thicker will lose less capacitance with voltage.
Higher voltage rating doesn't necessarily mean anything. I suspect for many values, most low voltage caps are just the same as higher voltage. I'm using random values as example: if they can make a 10nF 25V cap and it's already the minimum thickness to make sure it's not going to be too fragile and break while soldering, then all the lower voltage versions are probably going to be the exact same cap with a different label.
So if your problem is your DC-DC wants 10 real µF on the output but your 10µF cap only does 3µF at the voltage it's biased at, and you don't want to change the footprint and redo the board, you can put a higher voltage in the search but looking at the "thickness" info will save some time. Also using a thicker cap won't add inductance, whereas picking a larger package will.
Or just put more of them in parallel, which gives more capacitance and less inductance.
All the curves for Murata are available
This does not apply to C0G/NP0 which is an extremely stable and low distortion dielectric. But it has low K so you don't get µF values.
(*) not so random according to the link above, since there is hysteresis.