Properness for scaled gauged maps

We give an algebraic proof of properness of moduli stacks of gauged maps satisfying a stability conditition introduced by Mundet and Schmitt. The proof combines a git construction of Schmitt, properness for twisted stable maps by Abramovich-Vistoli, a variation of a boundedness argument due to Ciocan-Fontanine-Kim-Maulik, and a removal of singularities for bundles on surfaces in Colliot-Th\'el\`ene-Sansuc.
Comment: This was originally intended to be part of a survey for the AMS Summer Institute Proceedings, but the proof of properness grew too long and technical. The survey part [arXiv:1606.01384] was split off at the request of the editors, and is titled "Stable gauged maps". There is still substantial overlap between the two papers. This version has minor corrections