system-bath coupling. The dynamics of this class of models can be studied exactly, and it will help us to clarify several features of the interplay between non-adiabatic effects and incoherent transitions due to the external bath. In order to understand to which extent the results we find can be applied to more realistic cases, later we will consider a spin-$1/2$ one-dimensional Ising model. In such case, for the incoherent spin decay/pumping, the master equation cannot be mapped into local fermionic operators, therefore we will resort to a numerical study based on a matrix-product-operator (MPO) representation of the density matrix [@Verstraete_2004; @Zwolak_2004]. As we will discuss in more details in the rest of the paper, the overall phenomenology remains unchanged, thus reinforcing the fact that the exactly solvable models introduced here can be very useful benchmarks. We should remind that, in general, the thermodynamic properties of low-dimensional systems can be strongly affected by the dimensionality: for example, thermal fluctuations wash out quantum fluctuations of finite-temperature systems in one dimension, but not anymore in two dimensions. Furthermore, methods that are very powerful in one dimension, as the Jordan-Wigner transformation, are not applicable in higher dimensions. However, in the special case of free 