Some indications of how this may be studied are given in Andy Tonks' thesis in 1993 from Bangor. He assumes the spaces are represented by crossed complexes and shows how to calculate the homotopy colimit algebraically. This will not answer your question completely but may help. If the spaces $D(c)$ are Elienberg-Mac Lane spaces then there are interpretations of the idea using `higher generation by subgroups' in the work of Abels and Holz. There are also links with complexes of groups in a similar context.

Added later : the point about crossed complexes is that they ar every near to the chains on the universal covering of a space, but have a bit more homotopy recorded in them. The whole question of the generalisations of the van Kampen theorem come into this. You can find a lot on this in *R. Brown, P. J. Higgins and R. Sivera, 2010, Nonabelian Algebraic Topology: Filtered spaces,
crossed complexes, cubical homotopy groupoids , volume 15 of EMS Tracts in Mathematics ,
European Mathematical Society.* It includes discussions of Tonks results. For the simple case of a homotopy pushout a paper (R. Brown, E. J. Moore, T. Porter and C. D. Wensley, Crossed complexes, and free crossed
resolutions for amalgamated sums and HNN-extensions of groups , Georgian Math. J., 9,
(2002), 623 – 644) may be of interest.