This paper develops the unsteady continuous adjoint method for aeroacoustic problems governed by time-dependent turbulent flows. To predict flow-induced sound radiation from a body in free-stream, an incompressible Improved Delayed Detached Eddy Simulation is firstly performed. The generated noise is then propagated with the Kirchhoff Integral method, which uses the pressure distri- bution on the body surface to compute the sound pressure at selected receivers. The continuous adjoint method for the aforementioned process is developed and presented for the first time. In the adjoint process, the differentiated Kirchhoff Integral is used to compute the boundary condition of the adjoint velocity on the body surface and, then, the unsteady adjoint equations are solved backwards in time. It should be noted that the time window where the simulation is per- formed and the time window over which the objective function is evaluated do not coincide. This is reflected on the adjoint boundary condition along the body and the time integration of the sensitivity derivatives. Furthermore, to ensure the consistency of the continuous adjoint-based gradients, grid sensitivities are taken into account. By incorporating the grid displacement equations during the mathematical development of the adjoint method, the surface integral of the residual of the flow PDEs, which is commonly omitted, is replaced with a volume integral of the grid sensitivities and gives rise to the adjoint grid displacement equations along with an accurate sensitivity derivatives expression. The proposed method is verified against Finite Differences on a 3D turbulent flow around a cylinder and, then, applied to a real-world test case, concerned with the flow-induced sound radiation of the side mirror of the generic SAE vehicle.