  We introduce in this paper the numerical analysis of high order both in time
and space Lagrange-Galerkin methods for the conservative formulation of the
advection-diffusion equation. As time discretization scheme we consider the
Backward Differentiation Formulas up to order $q=5$. The development and
analysis of the methods are performed in the framework of time evolving finite
elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical
Analysis \textbf{41}, 1696-1845 (2021). The error estimates show through their
dependence on the parameters of the equation the existence of different regimes
in the behavior of the numerical solution; namely, in the diffusive regime,
that is, when the diffusion parameter $\mu$ is large, the error is
$O(h^{k+1}+\Delta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the
convergence is $O(\min (h^{k},\frac{h^{k+1} }{\Delta t})+\Delta t^{q})$. It is
worth remarking that the error constant does not have exponential $\mu ^{-1}$
dependence.
