  The aim of this article is to prove that diffusion processes in
$\mathbb{R}^d$ with a drift can be approximated by suitable Markov chains on
$n^{-1}\mathbb{Z}^d$. Moreover, we investigate sufficient conditions on the
conductances which guarantee convergence of the associated Markov chains to
such Markov processes. Analogous questions are answered for a large class of
nonsymmetric jump processes. The proofs of our results rely on regularity
estimates for weak solutions to the corresponding nonsymmetric parabolic
equations and Dirichlet form techniques.
