This paper continues our investigation into the question of when a homotopy $��= \{��_t\}_{t \in [0,1]}$ of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an isomorphism of the $K$-theory groups of the twisted groupoid $C^*$-algebras: $K_*(C^*(\mathcal{G}, ��_0)) \cong K_*(C^*(\mathcal{G}, ��_1)).$ In particular, we build on work by Kumjian, Pask, and Sims to show that if $\mathcal{G} = \mathcal{G}_��$ is the infinite path groupoid associated to a row-finite higher-rank graph $��$ with no sources, and $\{c_t\}_{t \in [0,1]}$ is a homotopy of 2-cocycles on $��$, then $K_*(C^*(\mathcal{G}_��, ��_{c_0})) \cong K_*(C^*(\mathcal{G}_��, ��_{c_1})),$ where $��_{c_t}$ denotes the 2-cocycle on $\mathcal{G}_��$ associated to the 2-cocycle $c_t$ on $��$. We also prove a technical result (Theorem 3.3), namely that a homotopy of 2-cocycles on a locally compact Hausdorff groupoid $\mathcal{G}$ gives rise to an upper semi-continuous $C^*$-bundle.

v2: Exposition condensed. This version to appear in the Pacific Journal of Mathematics