  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.
