Let $\mathcal{D}: C\to Top$ be a diagram of spaces (spaces are "nice", and $C$ is small). Let $X$ denote the homotopy colimit of $\mathcal{D}$ (which is connected) and $\pi(C)$ be the free groupoid on $C$ (i.e., fundamental groupoid of the geometric realization). 

Is it possible to characterize (or express) connected covers of $X$ in terms of (connected) groupoid covers of $\pi(C)$?

For example, if $C$ is acyclic and $\mathcal{D}$ assigns points then it can be done using the Groethendieck construction. 

Is there a general recipe that explains such a construction? 
Can it be done for diagrams of categories ?