There is by now 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 L\'evy 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. In regard to strictly M1 convergent moving averages and correlated CTRWs, it turns out that the convergence of stochastic integrals can fail markedly. Nevertheless, we are able to identify natural classes of integrand processes for which M1 convergence holds. We show that these results are general enough to yield functional limit theorems, in the M1 topology, for certain stochastic delay differential equations driven by moving averages.

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