This article analyses the transient magnetohydrodynamic free convective flow in vertical micro-concentric annuli in the presence of velocity slip and temperature jump at the outer surface of the inner cylinder and the inner surface of the outer cylinder. The Laplace transform technique has been used to find the solutions for the velocity and temperature fields by solving the governing partial differential equations in Laplace domain. However, the Riemann-sum approximation method is used to invert the Laplace domain to the time domain. The solution derived is validated by assenting comparison with exact solutions derived for the steady state which has been derived separately. An excellent agreement was found for transient and steady state at large value of time. The solution obtained for the velocity has been used to compute the skin friction, while the temperature has been used to compute the Nusselt number. The effect of various flow parameters entering into the problem such as time, Prandtl number, curvature radius ratio, Hartmann number, rarefaction parameter, and fluid–wall interaction parameter are discussed with the aid of line graphs.