We study the dipole structure of an extended redshift sample of Abell/ACO clusters in order to infer the dynamical origin of the motion of the Local Group (LG). To elucidate further the constraints that this motion places on dark matter models, we use numerical simulations based on an optimized version of the truncated Zel’dovich approximation which we have shown in an accompanying paper to provide a reliable representation of large-scale gravitational clustering. Taking advantage of their low computational cost, we run 20 realizations of each of six different dark matter models: four of these have a density parameter Ω 0 = 1, while the other two have Ω 0= 0.2, one with and one without a cosmological constant term. For the Abell/ACO sample, we have evaluated the parameter (where bcl is the linear bias parameter for the clusters), which reaches its asymptotic value at . Convergence occurs at this scale whether calculations are performed in the β LGor in the CMB frame, but the asymptotic value differs in these two cases: we find βLG=0.15 ± 0.04 and βCMB = 0.25 ± 0.06, respectively. After identifying in the simulations those observers having local densities and peculiar velocities similar to those of the Local Group, we construct mock cluster samples around them reproducing the same observational biases, and apply to these mock samples the same method of analysis as we used for the Abell/ACO sample. We find that an alignment between the cluster dipole and observer velocity (‘CMB’ dipole) directions, such as that observed (∆θ≲ 20°), should not be expected necessarily: much larger misalignment angles are often found in all models considered. This, together with the large observer-to-observer variance estimates of β, makes it difficult to place any firm constraints on cosmological models. This result suggests that the dipole analysis of the cluster distribution has a relevance that is cosmographical, rather than cosmological. Our results also demonstrate that the large amplitude and convergence depth of the observed cluster dipole cannot be taken as strong evidence either for or against a lowdensity Universe.