Summary: This study examines an initial-boundary value problem involving the system \[ \begin{cases} u_t = \Delta(u^m\phi(v)), \\ v_t = \Delta v - uv. \end{cases}\tag{\(\star\)} \] in a smoothly bounded domain \(\Omega\subset\mathbb{R}^n\) with no-flux boundary conditions, where \(m, n \geq 1\). The motility function \(\phi\in C^0([0, \infty))\cap C^3((0, \infty))\) is positive on \((0, \infty)\) and satisfies \[ \liminf_{\xi\searrow0} \frac{\phi(\xi)}{\xi^\alpha}>0 \quad\text{and}\quad \limsup_{\xi\searrow0}\frac{|\phi^\prime(\xi)|}{\xi^{\alpha - 1}} < \infty, \] for some \(\alpha > 0\). Through distinct approaches, we establish that, for sufficiently regular initial data, in two- and higher-dimensional contexts, if \(\alpha\in[1, 2m)\), then \((\star)\) possesses global weak solutions, while in one-dimensional settings, the same conclusion holds for \(\alpha > 0\), and notably, the solution remains uniformly bounded when \(\alpha \geq 1\). Furthermore, for the one-dimensional case where \(\alpha \geq 1\), the bounded solution additionally possesses the convergence property that \[ u(\cdot, t)\overset{\ast}{\rightharpoonup} u_\infty \text{ in }L^\infty(\Omega) \text{ and } v(\cdot, t)\rightarrow 0 \text{ in }W^{1, \infty}(\Omega)\text{ as } t\rightarrow\infty, \] with \(u_\infty\in L^\infty(\Omega)\). Further conditions on the initial data enable the identification of admissible initial data for which \(u_\infty\) exhibits spatial heterogeneity.