In the context of a microscopic approach to phenomenological irreversible thermodynamics, based upon nonequilibrium mechano-statistical foundations, we consider here questions related to the response of many-body nonequilibrium systems to thermal perturbations arising out of inhomogenities in the medium, We present a general theory of the resulting transport phenomena which is nonlocal in space and memory dependent, The limit of the local in space and instantaneous in time approximations is also considered and discussed. Propagation of damped waves is evidenced in equations of the Maxwell-Cattaneo type, which are generalizations of the diffusion-like equations of classical irreversible thermodynamics, A particular example of application of the theory is presented in the follow up article.2214478494