I once heard a rumour that various nice categories of stacks were co-complete. Gepner and Henriques, working from the groupoids point of view, give a construction [link] of 2-colimits of topological groupoids, but I haven't seen any definitive statement on whether there is a construction that works for, eg., differential or holomorphic stacks (let alone a proof). Personally I don't expect it, but I just want a definitive answer, so that I know that certain constructions I'm working on aren't a waste of time.
So what I'm asking is: are differential stacks co-complete? How about holomorphic stacks? If so, are there easy constructions? (Groupoid point of view preferred, but I'll take what I can get.) If the answer's no, are there any counterexamples? If none of these questions can be answered, can an expert come on here and say "it's not known"?