Context. The late evolutionary phase of low- and intermediate-mass stars is strongly constrained by their mass-loss rate, which is orders of magnitude higher than during the main sequence. The wind surrounding these cool expanded stars frequently shows nonspherical symmetry, which is thought to be due to an unseen companion orbiting the donor star. The imprints left in the outflow carry information about the companion and also the launching mechanism of these dust-driven winds. Aims. We study the morphology of the circumbinary envelope and identify the conditions of formation of a wind-captured disk around the companion. Long-term orbital changes induced by mass loss and mass transfer to the secondary are also investigated. We pay particular attention to oxygen-rich, that is slowly accelerating, outflows in order to look for systematic differences between the dynamics of the wind around carbon and oxygen-rich asymptotic giant branch (AGB) stars. Methods. We present a model based on a parametrized wind acceleration and a reduced number of dimensionless parameters to connect the wind morphology to the properties of the underlying binary system. Thanks to the high performance code MPI-AMRVAC, we ran an extensive set of 72 three-dimensional hydrodynamics simulations of a progressively accelerating wind propagating in the Roche potential of a mass-losing evolved star in orbit with a main sequence companion. The highly adaptive mesh refinement that we used, enabled us to resolve the flow structure both in the immediate vicinity of the secondary, where bow shocks, outflows, and wind-captured disks form, and up to 40 orbital separations, where spiral arms, arcs, and equatorial density enhancements develop. Results. When the companion is deeply engulfed in the wind, the lower terminal wind speeds and more progressive wind acceleration around oxygen-rich AGB stars make them more prone than carbon-rich AGB stars to display more disturbed outflows, a disk-like structure around the companion, and a wind concentrated in the orbital plane. In these configurations, a large fraction of the wind is captured by the companion, which leads to a significant shrinking of the orbit over the mass-loss timescale, if the donor star is at least a few times more massive than its companion. In the other cases, an increase of the orbital separation is to be expected, though at a rate lower than the mass-loss rate of the donor star. Provided the companion has a mass of at least a tenth of the mass of the donor star, it can compress the wind in the orbital plane up to large distances. Conclusions. The grid of models that we computed covers a wide scope of configurations: We vary the terminal wind speed relative to the orbital speed, the extension of the dust condensation region around the cool evolved star relative to the orbital separation, and the mass ratio, and we consider a carbon-rich and an oxygen-rich donor star. It provides a convenient frame of reference to interpret high-resolution maps of the outflows surrounding cool evolved stars.