  We study the velocity of the propagation of information for a class of local
dissipative quantum dynamics. This finite velocity is expressed by the
so-called Lieb-Robinson bound. Besides the properties of the already studied
dynamics, we consider an additional relation that expresses the propagation of
certain subspaces. The previously derived bounds did not reflect the
dissipative character of the dynamics and yielded the same result as for the
reversible case. In this article, we show that for this class the velocity of
propagation of information is time dependent and decays in time towards a
smaller velocity. In some cases the velocity becomes zero. At the end of the
article, the exponential clustering theorem of general frustration free local
Markovian dynamics is revisited.
