  We investigate the pseudoscalar (PS) meson ($\pi$ and $K$) quasi-distribution
amplitude (QDA), which is supposed to be an asymptotic analog to the meson
distribution amplitude (DA) $\phi_{\pi,K}(x)$ in the limit of the large
longitudinal PS-meson momentum, i.e. $p_3\to\infty$, in the nonperturbative
region. For this purpose, we employ the nonlocal chiral-quark model (NLChQM) in
the light-front formalism with a minimal Fock-state for the mesons $\sim
q\bar{q}$ at the low-energy scale parameter of the model $\Lambda\sim1$ GeV. As
a trial, we extract the transverse-momentum distribution amplitude (TMDA) from
the light-front wave function within the model, and convert it to QDA with help
of the virtuality-distribution amplitude. By doing that, we derive an
analytical expression for the nonperturbative QDA with the current-quark mass
correction up to $\mathcal{O}(\Delta m_q)$. Numerically, we confirm that the
obtained TMDA reproduces the experimental data for the photon-pion transition
form factor $F_{\gamma\gamma^*\pi^0}(Q^2)$ at the low-$Q^2$ qualitatively well.
We also observe that the obtained QDA approaches to DA as $p_3$ increases,
showing the symmetric and asymmetric curves with respect to $x$ for the pion
and kaon, respectively, due to the current-quark mass difference $m_{u,d}\ll
m_s$. Assigning $\xi\equiv2x-1$, the moments $\langle\xi^n\rangle_{\pi,K}$ are
computed, using the pion and kaon QDAs, and there appear only a few percent
deviations in the moments for $p_3\gtrsim30\Lambda$ in comparison to the values
calculated directly from DAs. It turns out that the higher moments are more
sensitive to the change of $p_3$, whereas the lower ones depend less on it.
