The Graetz problem in a transpiration-cooled channel was analytically attacked so as to explore the developing temperature field due to a sudden change in wall temperature of the channel subject to an arbitrary distribution of the local mass flux over the porous wall. Analytical expressions for the developments of the thermal boundary layer thickness, wall temperature, and Nusselt number were obtained for the thermal entrance region, assuming hydrodynamically forced convective flow in a channel with a locally variable blowing mass flux. When the blowing mass flux is kept constant over the wall surface, the cooling by the coolant is less effective near the entrance, thus, exposing to danger of thermal damage. This study reveals that the blowing mass flux graded inversely proportional to one-third power of the axial distance is quite effective to keep the wall temperature uniform. Numerical calculations based on finite volume method were also carried out to verify the analysis. The findings from this study can be applied to possible thermal managements of heat generating stacks such as in EV batteries and PEMFC, in which temperature uniformity is essential for product longevity.