  In the first part of the present paper, we show that strong convergence of
$(v_{0 \varepsilon})_{\varepsilon \in (0, 1)}$ in $L^1(\Omega)$ and weak
convergence of $(f_{\varepsilon})_{\varepsilon \in (0, 1)}$ in
$L_{\textrm{loc}}^1(\overline \Omega \times [0, \infty))$ not only suffice to
conclude that solutions to the initial boundary value problem \begin{align*}
  \begin{cases}
  v_{\varepsilon t} = \Delta v_\varepsilon + f_\varepsilon(x, t) & \text{in
$\Omega \times (0, \infty)$}, \\
  \partial_\nu v_\varepsilon = 0 & \text{on $\partial \Omega \times (0,
\infty)$}, \\
  v_\varepsilon(\cdot, 0) = v_{0 \varepsilon} & \text{in $\Omega$},
  \end{cases} \end{align*} which we consider in smooth, bounded domains
$\Omega$, converge to the unique weak solution of the limit problem, but that
also certain weighted gradients of $v_\varepsilon$ converge strongly in
$L_{\textrm{loc}}^2(\overline \Omega \times [0, \infty))$ 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 = \Delta u - \chi \nabla \cdot (\frac{u}{v} \nabla v) + g(u), \\
  v_t = \Delta v - uv,
  \end{cases} \end{align*} where $\chi > 0$ 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).
