  In this paper we derive an extra class of non-Markovian master equations
where the system state is written as a sum of auxiliary matrixes whose
evolution involve Lindblad contributions with local coupling between all of
them, resembling the structure of a classical rate equation. The system
dynamics may develops strong non-local effects such as the dependence of the
stationary properties with the system initialization. These equations are
derived from alternative microscopic interactions, such as complex environments
described in a generalized Born-Markov approximation and tripartite
system-environment interactions, where extra unobserved degrees of freedom
mediates the entanglement between the system and a Markovian reservoir.
Conditions that guarantees the completely positive condition of the solution
map are found. Quantum stochastic processes that recover the system dynamics in
average are formulated. We exemplify our results by analyzing the dynamical
action of non-trivial structured dephasing and depolarizing reservoirs over a
single qubit.
