The convergence of stochastic integrals driven by a sequence of Wiener processes $W_n\to W$ (with convergence in $C_t$) is crucial in the analysis of SPDEs. In this talk I shall focus on convergence of stochastic integrals of the form $\int_0^T V_n\, {\rm d} W_n \to \int_0^T V\,{\rm d} W$. Standard methods do not directly apply when $V_n$ converges to $V$ only weakly in the temporal variable. I shall discuss several convergence results that address the need to take limits of stochastic integrals when strong temporal convergence is absent. The key ingredient is an additional condition in the form of a uniform $L^1$ time translation estimate that is often available in SPDE settings but in itself insufficient to imply strong temporal compactness. This discussion will be in the context of applications to semilinear stochastic transport equations and stochastic conservation laws.