The paper presents a systematic framework for constructing generic models of nonequilibrium quantum dynamics that relax to a target stationary (mixed) state. The framework identifies all combinations of Hamiltonian and dissipative dynamics that achieve the desired steady state, generalizing Davies' generator for finite-temperature relaxation to nonequilibrium dynamics targeting arbitrary stationary states. The focus is on Gibbs states of stabilizer Hamiltonians, and the formalism constrains the rates of dissipative and unitary processes to find local Lindbladians compatible with the target state. The methods also identify operations, including syndrome measurements and local feedback, to correct errors in Lindbladians that are not compatible with the target state. The framework reveals new models of quantum dynamics, such as a "measurement-induced phase transition" where two-point functions exhibit critical scaling with distance. Time-reversal symmetry, defined within the formalism, can be broken in both classical and quantum ways. The framework provides a starting point for exploring dynamical universality classes in open quantum systems and identifying protocols for quantum error correction.The paper presents a systematic framework for constructing generic models of nonequilibrium quantum dynamics that relax to a target stationary (mixed) state. The framework identifies all combinations of Hamiltonian and dissipative dynamics that achieve the desired steady state, generalizing Davies' generator for finite-temperature relaxation to nonequilibrium dynamics targeting arbitrary stationary states. The focus is on Gibbs states of stabilizer Hamiltonians, and the formalism constrains the rates of dissipative and unitary processes to find local Lindbladians compatible with the target state. The methods also identify operations, including syndrome measurements and local feedback, to correct errors in Lindbladians that are not compatible with the target state. The framework reveals new models of quantum dynamics, such as a "measurement-induced phase transition" where two-point functions exhibit critical scaling with distance. Time-reversal symmetry, defined within the formalism, can be broken in both classical and quantum ways. The framework provides a starting point for exploring dynamical universality classes in open quantum systems and identifying protocols for quantum error correction.