The authors investigate systematically the role of upstream and downstream boundary conditions on the heat (or mass) transfer when axial diffusion is effective. As examples, three similar cases of plug flow in channels flows are studied. In all cases there is a central heating section and the fluid flows from there into an insulated semi-infinite conduit. The situations prevailing upstream of this section vary. As expected it is found that these situations have a substantial influence when the Péclét number is low, and the heating section is short. The analysis is carried out assuming a uniform profile. It is, nevertheless, less than straightforward because in all three cases the boundary conditions are mixed. Solutions are obtained in the form of Fourier Integrals the inversion of which is carried out numerically.