  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 $4$.
