Don't be fooled into thinking I'm on some paper-finishing spree (I'm not!), but there is another piece of writing you can read: Projectivity of good moduli spaces of semistable quiver representations and vector bundles.
2 weeks ago there was an Oberwolfach workshop on algebraic geometry and noncommutative algebra, at which I spoke about upcoming work with Chiara Damiolini, Hans Franzen, Vicky Hoskins, Sveta Makarova and Tuomas Tajakka on redoing the construction of moduli spaces of semistable quiver representations in the language of algebraic stacks, rather than the original construction using geometric invariant theory. This question was in fact directly inspired by an expository paper written together with Jarod Alper, Daniel Bragg, Jason Liang and Tuomas Tajakka for the analogous question for moduli of vector bundles, where such a construction was known (albeit not in the language of algebraic stacks and their good moduli spaces, which are more recent) due to Faltings.
The full paper will be finished soon (but this is a very flexible notion), so for now you'll have to do with an extended abstract that contains more a discussion of how various constructions for curves and quivers are very similar if you look at them in the right way, rather than the details for the proofs.
Maybe we should arXiv the expository paper for moduli of vector bundles too. The print version will be available by the end of the year by the way.