  Currents through quantum systems may probe non-analyticities in
quantum-critical many-body ground states. For a large class of dissipative
quantum critical systems we show that it is possible to obtain the reduced
system dynamics in the vicinity of quantum critical points in a
thermodynamically consistent way, while capturing non-Markovian effects. We
achieve this by combining reaction coordinate mappings with polaron transforms.
Exemplarily, we consider the Lipkin-Meshkov-Glick model in a transport setup,
where the quantum phase transition manifests itself in the heat transfer
statistics.
