Is capacitance constant?
Yes, capacitance, for a given capacitor is strictly a constant. It doesn't depend on the EMF of the charging source or on the charges at the plates at some given instant.
The capacitance, depends on two factors. The first and foremost one is, the geometry of the plates which includes size, separation, relative orientation and shape of the plates.
The second factor is the permittivity of the material medium intervening between the plates (i.e. to the dielectric constant/ relative permittivity of the medium).
The capacitance increases by a factor of the dielectric constant (of the medium) if such a medium totally occupies the region between the plates.
This happens due to polarized fields within the dielectric which tend to partially cancel the effect of the electric field between the plates and thus, the potential.
The charge stored remains same and thus, one can infer that, the capacitance has increased. (i.e. more charge can be stored for the same difference in potential between the two points.)