We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly-varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. In particular, we recover the zero temperature many-body adiabatic theorem by first taking the thermodynamic limit and then the zero temperature limit. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation and fermionic cluster expansion. This talk is based on a joint work with R. L. Greenblatt, M. Lange and M. Porta.