We provide criteria for Itô integration to behave continuously with respect to Skorokhod’s \(J_1\) and \(M_1\) topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the \(M_1\) setting and unify existing theories in the \(J_1\) case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, \(M_1\) tightness in fact implies \(J_1\) tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak \(M_1\) and \(J_1\) convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.
