  We investigate the non-relativistic limit of the Klein--Gordon equation for
mixed scalar particles and show that, in this regime, one unavoidably arrives
at redefining the particle's inertial mass. This happens because, in contrast
to the case when mixing is absent, the antiparticle sector contribution cannot
be neglected for particles with definite flavor. To clearly demonstrate this
feature, we adopt the Feshbach--Villars formalism for Klein--Gordon particles.
Furthermore, within the same framework, we also demonstrate that, in the
presence of a weak gravitational field, the mass parameter that couples to
gravity (gravitational mass) does not match the effective inertial mass. This,
in turn, implies a violation of the weak equivalence principle. Finally, we
prove that the Bargmann's superselection rule, which prohibits oscillating
particles on the basis of the Galilean transformation, is incompatible with the
non-relativistic limit of the Lorentz transformation and hence does not collide
with the results obtained.
