Shock tracking aims to generate a mesh such that element faces align with shock surfaces and other non-smooth features to perfectly represent them with the inter- element jumps in the solution basis, e.g., in the context of a finite volume or discontinuous Galerkin (DG) discretization. These methods lead to high-order approximations of high-speed flows and do not require nonlinear stabilization or extensive refinement in non-smooth regions because, once the non-smooth features are tracked by the mesh, the high-order solution basis approximates the remaining smooth features. High-Order Implicit Shock Tracking (HOIST) recasts the geometrically complex problem of generating a mesh that conforms to all discontinuity surfaces as a PDE-constrained optimization problem. The optimization problem seeks to determine the flow solution and nodal coordinates of the mesh that simultaneously minimize an error-based indicator function and satisfy the discrete flow equations. A DG discretization of the governing equations is used as the PDE constraint to equip the discretization with desirable properties: conservation, stability, and high-order accuracy. By using high-order elements, curved meshes are obtained that track curved shock surfaces to high-order accuracy. The optimization problem is solved using a sequential quadratic programming method that simultaneously converges the mesh and DG solution, which is critical to avoid nonlinear stability issues that would come from computing a DG solution on an unconverged (non-aligned) mesh. In this work, the HOIST method is further extended to simulate time-dependent, inviscid flows for higher dimensions. We use a space-time formulation of the governing equations and perform shock tracking over a space-time slab. In the D-dimensional space and time setting we generate a (D+1)-dimensional mesh by extruding a spatial mesh of D-dimensional hypercube elements into (D+1)-dimensional hypercube elements. We also introduce robustness measures for general hypercube element edge collapses, as well as a method to leverage information from previous time-slabs to improve the initial guess for our mesh on the current slab.