  We consider an experiment to search for dark sector particles in dark photon
kinetic mixing model by analyze invisible and semi-invisible decays of neutral
mesons $M^0 = \pi^0$, $\eta$, $\eta'$, $\omega$, $f_2(1270)$, produced in the
NA64 experiment at the CERN SPS. The approach proposed in
Ref.~\cite{Gninenko:2014sxa} is to use the charge-exchange reactions $\pi^- +
(A, Z) \to M^0 + (A,Z-1); M^0 \to$ invisible or semi-invisible of high-energy
pions (or kaons) at a nuclei target as a source of $M^0$s, which subsequently
decay invisibly into dark sector. This reaction chain would lead to a striking
signature of the signal event - the complete disappearance of the beam energy
in the setup. Using data obtained from the study of charge-exchange reactions
at IHEP (Protvino) and Fermilab (Batavia) we show that the integral cross
sections $\sigma$ for production of the neutral mesons $M^0$ are slightly
deviate from phenomenological formula $\sigma \sim Z^{2/3}$, where $Z$ is the
nuclei charge. In particular, we present the formulas for the differential and
integral sections that explicitly depend on the Mandelstam and $Z$ variables.
Derived formulas are used to predict the cross sections as a function of beam
energy for several target nuclei, and to estimate the projection sensitivity
for the proposed search for the $M^0\to$ semi-invisible and $M^0\to$ invisible
decays through the vector portal to dark sector. Sensitivity to different
semi-invisible decay modes of neutral pseudoscalar mesons is studied.
