This paper aims to investigate the transfer of heat phenomenon in a hydromagnetic time dependent flow of micropolar fluid across an oscillating stretchable curved surface by using the Cattaneo–Christov heat flux model, which considers thermal relaxation time. An elastic curved surface that stretches back and forth causes the flow situation. The flow equations are derived as nonlinear partial differential equations by incorporating a curvilinear coordinates system, which is then solved analytically via the homotopy analysis method (HAM). The accuracy of the derived analytical results is also examined by using a finite-difference technique known as the Keller box method, and it is found to be in strong agreement. The influences of various physical characteristics such as material parameter, magnetic parameter, thermal relaxation parameter, a dimensionless radius of curvature, Prandtl number and ratio of surface’s oscillating frequency to its stretching rate parameter on angular velocity, fluid velocity, pressure, temperature, heat transmission rate, and skin friction and couple stress coefficient are depicted in detail with the help of graphs and tables. Furthermore, for the verification and validation of the current results, a tabular comparison of the published data in the literature for the case of flat oscillating surface versus curved oscillating surface is carried out and found to be in good agreement.