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In the first part of the present paper, we show that strong convergence of $(v_{0 \varepsilon})_{\varepsilon \in (0, 1)}$ in $L^1(Ω)$ and weak convergence of $(f_{\varepsilon})_{\varepsilon \in (0, 1)}$ in $L_{\textrm{loc}}^1(\overline Ω\times [0, \infty))$ not only suffice to conclude that solutions to the initial boundary value problem \begin{align*} \begin{cases} v_{\varepsilon t} = Δv_\varepsilon + f_\varepsilon(x, t) & \text{in $Ω\times (0, \infty)$}, \\ \partial_νv_\varepsilon = 0 & \text{on $\partial Ω\times (0, \infty)$}, \\ v_\varepsilon(\cdot, 0) = v_{0 \varepsilon} & \text{in $Ω$}, \end{cases} \end{align*} which we consider in smooth, bounded domains $Ω$, converge to the unique weak solution of the limit problem, but that also certain weighted gradients of $v_\varepsilon$ converge strongly in $L_{\textrm{loc}}^2(\overline Ω\times [0, \infty))$ along a subsequence. We then make use of these findings to obtain global generalized solutions to various cross-diffusive systems. Inter alia, we establish global generalized solvability of the system \begin{align*} \begin{cases} u_t = Δu - χ\nabla \cdot (\frac{u}{v} \nabla v) + g(u), \\ v_t = Δv - uv, \end{cases} \end{align*} where $χ> 0$ and $g \in C^1([0, \infty))$ are given, merely provided that ($g(0) \geq 0$ and) $-g$ grows superlinearily. This result holds in all space dimensions and does neither require any symmetry assumptions nor the smallness of certain parameters. Thereby, we expand on a corresponding result for quadratically growing $-g$ proved by Lankeit and Lankeit (Nonlinearity, 32(5):1569--1596, 2019).