In this work we will introduce and analyze the Arbitrary Lagrangian Eulerian formulation for a model problem of a scalar advection-diffusion equation defined on a moving domain. Moving from the results illustrated in our previous work [J. Num. Math. 7 (1999) 105], we will consider first and second-order time advancing schemes and analyze how the movement of the domain might affect accuracy and stability properties of the numerical schemes with respect to their counterpart on fixed domains. Theoretical and numerical results will be presented, showing that stability properties are not, in general, preserved, while accuracy is maintained.