Your are talking about action of a groupoid on a space via homeomorphisms. Let $G$ be a groupoid. Then, in general, one may replace the category $\mathcal{Top}$ by a bicategory $\mathcal{A}$ and talk about an action of $G$ on $\mathcal{A}$. A good reference for this purpose is the following paper of Meyer, Buss and Zhu: https://arxiv.org/pdf/0908.0455v1.pdf wherein they define and study actions of group(oid)s on bicategories.

An interesting case of these actions is when $\mathcal{A}$ is the bicategory  of $C^*$-algebras, say $\mathcal{C}^*$. In $\mathcal{C}^*$, the 'objects' are $C^*$-algebras, a 'functor' or 1-arrow from a $C^*$-algebra $A$ to $B$ is a $C^*$-correspondence from $A$ to $B$, and a 2-arrow is an isomorphism of $C^*$-correspondences. Meyer, Buss and Zhu show that, in discrete cases, these actions are same as equivalent to saturated Fell bundles on groupoids. This result is true when $G$ is a topological group. Thus your question has a significant meaning in some other category.

Moving back to your example, Meyer-Buss-Zhu's work shows that $\mathcal{Top}$ is kind of smaller category to work with. You may form the (bi)category $\mathcal{Q}$ of topological quivers introduced by Muhly and Tomford : http://arxiv.org/abs/math/0312109. Then and action of $G$ on on $\mathcal{Q}$ gives, what we may call, a `topological' version of Fell bundles, $B\to G$. Let $B_{\eta}$ denote the fibre on $\eta\in G$. 
If $G$ acts on the subcategory $\mathcal{Top}\subseteq \mathcal{Q}$, then the equivalence between $B_{s(\gamma)}$ and $B_{r(\gamma)}$ is just a homeomorphism.