Solving PDEs on time-varying domains has many applications in computational fluid dynamics. Generally, one has to discretize the PDE and track the variation (movement + deformation) of the domain simultaneously. We propose a high-order numerical method for solving linear convection-diffusion equations in 2D based on fictitious domain and Eulerian meshes. For smooth solutions, high-order error estimates are proved by taking account of surface-tracking errors, time-discretization errors, and spatial errors from unfitted finite element discretization. Numerical experiments show up to fourth-order convergence of the method for relatively large deformation of the domain.