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 Lévy-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 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.