We develop high-order numerical methods for solving advection-diffusion equation and the Oseen equation (linearized Navier-Stokes equations) with time-moving interfaces. The methods are based on unfitted finite element discretization on fixed Eulerian meshes of the domain. The locus of the moving interface is tracked by high-order cubic MARS algorithms. We present thorough error estimates for the methods (orders 2≤k≤4) by taking full consideration of all errors from interface-tracking, spatial discretization, and temporal integration. Numerical experiments demonstrate the optimal convergence of the methods for k = 3 and 4.