We characterise the convergence of a certain class of discrete time Markov
processes toward locally Feller processes in terms of convergence of associated
operators. The theory of locally Feller processes is applied to L\'evy-type
processes in order to obtain convergence results on discrete and continuous
time indexed processes, simulation methods and Euler schemes. We also apply the
same theory to a slightly different situation, in order to get results of
convergence of diffusions or random walks toward singular diffusions. As a
consequence we deduce the convergence of random walks in random medium toward
diffusions in random potential