The Abramovich-Vistoli/Chen-Ruan theory of twisted stable maps into Deligne-Mumford stacks is extremely useful, as is the generalization to tame Artin stacks in positive characteristic. I am wondering if there are any useful analogues of this theory (say in characteristic 0) for maps into Artin stacks with infinite stabilizers.

Obviously some strong restrictions must apply. For instance, maps from curves into the classifying stack $B\mathbb{C}^*$ of $\mathbb{C}^*$ correspond to line bundles on the source curve, and so the stack of these maps cannot be separated without some strong condition.

Has anyone thought about what could be done to build a useful theory of stable maps into some types of Artin stack with infinite stabilizers? Are there some conditions on the target stack so that the resulting stack of stable maps, whatever those are, would form a nice stack itself: say, proper? Deligne-Mumford? with projective coarse moduli space? Or are these things completely unreasonable to hope for?