Perhaps the most natural example is given by universal covers?

Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X_x$ be the universal covering of $X$ taken at $x$ (the fiber at $y \in X$ is the homotopy classes of paths in $X$ which start in $x$ and end in $y$, where we are taking homotopy classes relative to $\lbrace0,1\rbrace$).

Let $\pi$ be the fundamental groupoid of $X$. Then there is a functor $\pi\to \text{Top}$ given on objects by $x\mapsto \tilde X_x$. On morphisms of $\pi$ from $x$ to $y$, the functor is given by the map $\tilde X_x \to \tilde X_y$ that is induced by concatenating paths.

Perhaps the most natural example is given by universal covers?

Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X_x$ be the universal covering of $X$ taken at $x$ (the fiber at $y \in X$ is the homotopy classes of paths $[0,1]\to X$ which start at $x$ and end at $y$, where we are taking homotopy classes relative to $\lbrace0,1\rbrace$).

Let $\pi$ be the fundamental groupoid of $X$. Then there is a functor $\pi\to \text{Top}$ given on objects by $x\mapsto \tilde X_x$. On morphisms of $\pi$ from $x$ to $y$, the functor is given by the map $\tilde X_x \to \tilde X_y$ that is induced by concatenating paths.