When working with high-order methods it is well known that accuracy can be significantly affected when the domain boundary is curved. For this reason, special treatments of the boundary condition must be implemented to preserve the expected accuracy. This is particularly crucial for high-order methods, where errors due to geometric approximation can even exceed those related to the actual discretization of the scheme. In fact, high-order methods can be used on complex domains with unstructured meshes and can exhibit higher convergence rates only on properly constructed meshes. However, increased accuracy comes at a price. A significant computational cost in high-order simulations is the generation of unstructured curvilinear meshes: this is typically achieved by increasing the degrees of freedom per grid cell. The presentation will focus on recent developments in the field of high-order unfitted boundary conditions and their application to the field of hyperbolic equations for modeling compressible fluid flows on moving unstructured linear meshes. The new methods are expected to compensate the lack of precision due to the linear approximation of moving curved boundaries and have a relevant impact in the field of Lagrangian hydrodynamics and fluid-structure interaction.