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 ?



Connected covering spaces of a homotopy colimit
I have not quite thought this through but there are some facts which seem relevant:


1. For suitably nice $X$, the fundamental groupoid functor gives an equivalence of categories from covering maps over $X$, to covering groupoids over $\pi\_1X$; this is proved in my book "Topology and groupoids" (2006) and was in the (differently titled) 1988 edition, and in essence way back in the 1968 edition; it gives an essentially base point free approach to covering space theory. One advantage is also that a *map* of spaces is modelled by a *morphism* of groupoids, instead of an action. Lifting morphisms is easier to follow than lifting actions, at least for me.
2. If $p: Y \to X$ is a map in the category of compactly generated spaces, then the pullback functor $p^\*: Top/X \to Top/Y$ has a right adjoint and so preserves colimits. (This is one aspect of the exponential law in spaces over $X$, largely due to Peter Booth, see his entry on MathSciNet.)
3. If $q: H \to G$ is a fibration of groupoids, then the pullback functor $p^\*: Gpd/G \to Gpd/H$ has a right adjoint and so preserves colimits. (This is part of the restricted locally closed aspect of the category of groupoids, which is due to F. Conduch\'e, and even earlier to J. Giraud: in short, the middle morphism of the supposed exponential law has to be a fibration.)


Of course covering morphisms of groupoids are special cases of fibrations. 


I thought homotopy colimits are put together from various kinds of colimits; if so that ought to do the job. 


Arguments of this type are used in 


R. Brown and R. Street, `Covering morphisms of crossed
complexes and of cubical omega-groupoids are closed under tensor
product', Cah. Top. G\'eom. Diff. Cat., 52 (2011) 188-208.


