  This work considers the Keller-Segel consumption system \begin{eqnarray*}
\left\{ \begin{array}{llll} u_t=\Delta (u\phi(v))+au-bu^\gamma,\quad &x\in
\Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0 \end{array}
\right. \end{eqnarray*} in a smoothly bounded domain $\Omega\subset
\mathbb{R}^n,$ $n\geq1$, under no-flux boundary conditions, where the
parameters $a,b>0$, $\gamma\geq2$, and the motility function $\phi$ suitably
generalizes the prototype given by $\phi(s)=s^\alpha$ for all $s\geq0$ with
$\alpha>0$.
  When $\phi$ is appropriately smooth with $\alpha\geq1$, it is shown that if
one of the following cases holds: (i) $\gamma>2$; (ii) $\gamma=2$, either
$n\leq2$ or $n\geq3$ and $b$ is sufficiently large, then for all suitably
regular initial data global classical solutions can be constructed. Whereas
when $\phi$ is considered to be with rather mild regularity properties and
$\gamma=2$, for arbitrary $b>0$, this system admits at least one global weak
solution in case $\alpha>0$. In addition, if $\phi$ is suitably smooth with
$\alpha>1$, then the above weak solutions become eventually smooth.
