  In Klingenberg, Schn\"ucke and Xia (Math. Comp. 86 (2017), 1203-1232) an
arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve
conservation laws has been developed and analyzed. In this paper, the ALE-DG
method will be extended to several dimensions. The method will be designed for
simplex meshes. This will ensure that the method satisfies the geometric
conservation law, if the accuracy of the time integrator is not less than the
value of the spatial dimension. For the semi-discrete method the L2-stability
will be proven. Furthermore, an error estimate which provides the suboptimal
(k+1/2) convergence with respect to the L-infinity-norm will be presented, when
an arbitrary monotone flux is used and for each cell the approximating
functions are given by polynomials of degree $k$. The two dimensional
fully-discrete explicit method will be combined with the bound preserving
limiter developed by Zhang, Xia and Shu in (J. Sci. Comput. 50 (2012), 29-62).
This limiter does not affect the high order accuracy of a numerical method.
Then, for the ALE-DG method revised by the limiter the validity of a discrete
maximum principle will be proven. The numerical stability, robustness and
accuracy of the method will be shown by a variety of two dimensional
computational experiments on moving triangular meshes.
