The inter-grain conductivity (sigma(GB)) of polycrystalline and granular Sr2FeMoO6 (SFMO) increases linearly with temperature between liquid He to room temperature at least. This very unusual behavior motivated the new investigations reported here. It was found that sigma(GB)(T) is linear also for granular samples of La-doped SFMO and for sintered SFMO with modified grain boundaries (GBs), but it is nonlinear for granular samples of Ag- and Na-doped SFMO. The fluctuation-induced tunneling (FIT) model was successfully applied in the interpretation of both types of behavior. Current voltage measurements were carried out on a resistive sample with oxygenated GBs, using pulsed high currents. The interpretation of the effect of a high electric field on transport, in terms of the FIT model, is discussed. (C) 2004 Elsevier B.V. All rights reserved.