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Systems with local dynamics are characterized by a finite velocity ofpropagation of perturbations, known as the Lieb-Robinson velocity. On the otherhand, irreducible stochastic processes drive states towards some unique fixedpoint. However, combining both effects is mathematically challenging. Thebounds on propagation do not depend on system-size, while the theory of mixingis mostly based on extensive upper-bounds. In a previous paper, a class oflocal Lindbladian operators on arbitrary lattice was constructed, for which thetwo effects could be combined. In this paper, we show that for some localdynamics, local observables are propagated inside a much smaller convex subsetof the total Banach space. This allows us to show localization for suchdynamics.

Killing of Transport in Lattices driven by Local Quantum Stochastic Dynamics