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Given a sequence of positive Hermitian holomorphic line bundles $(L_p,h_p)$on a K\"ahler manifold $X$, we establish the asymptotic expansion of theBergman kernel of the space of global holomorphic sections of $L_p$, under anatural convergence assumption on the sequence of curvatures $c_1(L_p,h_p)$. Wethen apply this to study the asymptotic distribution of common zeros of randomsequences of $m$-tuples of sections of $L_p$ as $p\to\infty$.

Bergman kernels and equidistribution for sequences of line bundles on K\"ahler manifolds