The unsteady viscous flow and heat transfer in the vicinity of an axisymmetric stagnation point of an infinite moving plate with transpiration W0 are investigated when the axial velocity and wall temperature vary arbitrarily with time. The free stream is steady and with a strain rate of a. An exact solution of the Navier–Stokes equations and energy equation is derived in this problem. A reduction of these equations is obtained by the use of appropriate transformations for the most general case when the transpiration rate is also time-dependent, but results are presented only for uniform values of this quantity. The general self-similar solution is obtained when the axial velocity of the plate and its wall temperature vary as specified time-dependent functions. For completeness, sample semi-similar solutions of the unsteady Navier–Stokes equations have been obtained numerically using a finite difference scheme. All the solutions above are presented for different values of dimensionless transpiration rate, S = W0/√ aυ, where υ is the kinematic viscosity of the fluid. The effects of the sundry parameters, including transpiration rate, Prandtl number, oscillation frequency and accelerating/decelerating parameter, on the velocity and temperature profiles, as well as surface shear stresses and heat transfer coefficient, are investigated and results are shown through graphs.