PDE-constrained optimization has become an essential tool in the design of steady-state engineering systems, but is less commonly used for unsteady problems due to the large cost associated with repeated solving the underlying PDE. However, there is a large-class of problems that are inherently unsteady, such as flow past airfoils at a high angle of attack, flow through wind farms, and flapping flight, that would benefit from numerical optimization. In this work, a globally high-order numerical scheme is developed for the discretization of conservation laws on deforming domains, and the corresponding fully-discrete adjoint method is derived. The numerical scheme is used to compute relevant output functionals of the conservation law, and their gradients, for the use in gradient-based nonlinear optimization. An adaptive model reduction strategy is developed for reducing the cost of the numerical solver by searching for the solution in a small affine subspace, which is built on-the-fly during the optimization procedure. The proposed framework is demonstrated on a selection of optimal control and shape optimization problems, where the compressible Navier-Stokes equations are taken as the constraining PDE.