There is an extensive theory of weak convergence for moving averages and continuous-time random walks (CTRWs) with respect to Skorokhod's M1 and J1 topologies. Here we address the fundamental question of how this translates into functional limit theorems in the M1 or J1 topology for stochastic integrals driven by these processes. As an important application, we provide weak approximation results for general SDEs driven by time-changed Lévy processes. Such SDEs and their associated fractional Fokker--Planck--Kolmogorov equations are central to models of anomalous diffusion in statistical physics. Our results yield a rigorous functional characterisation of these as continuum limits of the underlying models driven by CTRWs. With regard to strictly M1 convergent moving averages and correlated CTRWs, it turns out that the convergence of stochastic integrals can fail decidedly and fundamental new challenges arise compared to the J1 setting. Nevertheless, we identify natural classes of integrand processes for which there is M1 convergence of the stochastic integrals. We also show that these results are flexible enough to yield functional limit theorems in the M1 topology for certain stochastic delay differential equations driven by moving averages.