This paper presents a theoretical analysis of blood flow and heat transfer in a permeable vessel in the presence of an external magnetic field. The unsteadiness in the coupled flow and temperature fields is considered to be caused due to the time-dependent stretching velocity and the surface temperature of the vessel. The non-uniform heat source/sink effect on blood flow and heat transfer is taken into account. This study is of potential value in the clinical treatment of cardiovascular disorders accompanied by accelerated circulation. The problem is treated mathematically by reducing it to a system of coupled nonlinear differential equations, which have been solved by using similarity transformation and boundary layer approximation. The resulting nonlinear coupled ordinary differential equations are solved numerically by using an implicit finite difference scheme. Computational results are obtained for the velocity, temperature, the skin-friction coefficient and the rate of heat transfer in the vessel. The estimated results are compared with another analytical study reported earlier in scientific literatures. The present study reveals that the heat transfer rate is enhanced as the value of the unsteadiness parameter increases, but it reduces as the space-dependence parameter for heat source/sink increases.