We develop a high-order finite element method to solve the advection-diffusion equation on a time-varying domain. The method is based on a characteristic-Galerkin formulation combined with the $k^{\rm th}$-order backward differentiation formula (BDF-$k$) and the fictitious-domain finite element method. Optimal error estimates of the discrete solutions are proven for $2\le k\le 4$ by taking account of the errors from interface-tracking, temporal discretization, and spatial discretization, provided that the $(k+1)^{\rm th}$-order Runge-Kutta scheme is used for interface-tracking. Numerical experiments demonstrate the optimal convergence of the method for $k=3$ and $4$.

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