  We prove the convergence of solutions of nonlocal conservation laws to their
local entropic counterpart for a fundamentally extended class of nonlocal
kernels when these kernels approach a Dirac distribution. The nonlocal kernels
are assumed to have fixed support and do not have to be monotonic. With sharp
estimates of the nonlocal kernels and a surrogate nonlocal quantity, we prove
compactness in $C(L^{1}_{\text{loc}})$ which allow passing to the limit in the
weak formulation. A careful analysis of the entropy condition of local
conservation laws together with the named estimators for the considered kernels
enable it to prove the entropy admissibility of the nonlocal equation in the
limit, completing the convergence proof.
