  Given a sequence of Hermitian holomorphic line bundles $(L_k,h_k)$ over a
complex manifold $M$ which may not be compact, we generalize the scaling method
in arXiv:2310.08048 to study the asymptotic behavior of the Bergman kernels and
spectral kernels with respect to the space of global holomorphic sections of
$L_k$ with $(0,q)$-forms. We derive the leading term of the Bergman and
spectral kernels under the local convergence assumption in the sequence of
Chern curvatures $c_1(L_k,h_k)$, inspired by arXiv:2012.12019. The manifold $M$
may be non-K\"ahler and $c_1(L_k,h_k)$ may be negative or degenerate. Moreover,
we establish the $L_k$-asymptotic version of Demailly's holomorphic Morse
inequalities as an application to compact complex manifolds.
