I will discuss some new Artin stack compactifications of the moduli of smooth curves, which parameterize "perturbed curves'" consisting of stacky semistable curves equipped with line bundles that are trivialized away from the rational bridges. This yields an infinite family of stacks all of which admit the classical coarse space of stable curves as a good moduli space. As an application, we construct a proper and flat moduli of Gieseker SL_n-bundles over the stack of stable curves, parameterizing SL_n-bundles on perturbed curves subject to a stability condition. The resulting moduli problem is integral, lci and has an explicit tangent obstruction theory; it could be viewed as a version of stable maps into the classifying stack BSL_n. This talk is based on part of joint work in progress with Damiolini, Halpern-Leistner, Inchiostro and Stephens.