The authors consider an Ising system in two dimensions with anisotropic ferromagnetic interactions in the strong anisotropic limit and study, via numerical simulation, the dynamics of the interface separating two domains. Since the system is highly anisotropic (Jx>>Jy with Jx>>kBT) and they neglect the overhang configurations, the model in some aspects is an SOS (solid on solid) model. In this case the domain wall moves in one direction (x) and they are in the so-called 'strip geometry' (L* infinity ), L being the size of the system in the y direction. The dynamics of this interface can be reduced, as has been already shown, to the correlated motion of random walkers. Their previous study at high temperature (Jy<kBT) has shown that for the equilibrium case where the mean position of the centre of mass (CM) does not change, the exponents z and alpha of the scaling relation describing the dynamics of the width of the wall have values 2 and 0.5, respectively. An equality z-2 alpha =1 was also obtained from cross-over arguments. In this paper they extend their study, by including a uniform external magnetic field, to the nonequilibrium case where CM mean position of the interface moves with time. They consider both the high- and low-temperature cases (Jy/kBT=0.1 and 1), and obtain the equality z-2 alpha =- alpha CM; alpha CM being exponent characterizing the size dependence of the diffusion coefficient of the CM, i.e. D approximately Lalpha CM in the long-time regime. For equilibrium they get alpha CMapproximately -1. For the low-temperature, field-driven case they find the exponent approaching the value -0.5 as the magnetic field increases from 0 to H/Jy=2. Since the static exponent alpha obtained is always near 0.5, their results in the low-temperature case correspond to z=2 for equilibrium and approach the value z=3/2 predicted by Kardar, Parisi and Zhang in the non-equilibrium situation. The values of the exponent z obtained in different cases (equilibrium and non-equilibrium) by calculating the CM exponent alpha CM are the same as those obtained from known equalities: z+ alpha =2 (non-equilibrium) and z-2 alpha =d-1 (equilibrium). Therefore they propose that the single equality z-2 alpha =- alpha CM may apply far more generally and the study of CM dynamics may therefore provide an alternative (or complementary) way of analysing the results of domain growth simulations. They also note that their results are in agreement with two-dimensional results on the restricted solid on solid model (RSOS).