Simulations of the turbulent cascade forming in the solar wind, including cross helicity, commonly adopt a homogeneous setup, not taking into account wind expansion. Here we want to assess the predictions of decaying 3D compressible (low Mach number) MHD simulations, respectively homogeneous and with expansion, in order to examine which is the most fruitful approach to understanding the turbulent cascade in the solar wind. We follow turbulent evolution during 10 nonlinear turnover times, considering several initial values of the initial spectral slope and cross helicity. In the expanding case, the transverse sizes of the plasma volume are stretched by a factor of 5 during the simulation, corresponding to traveling from 0.2 up to 1 au. In homogeneous simulations, the relative cross helicity rises, and the Elsässer spectra E ± show "pinning," with a steep dominant spectrum and flat subdominant spectrum, the final spectral indices depending on cross helicity but not initial indices. With expansion, the relative cross helicity decreases, and dominant and subdominant spectra share the same index, with the index relaxing to an asymptotic value that generally depends on the initial index. The absence of pinning, as well as the decrease of relative cross helicity, probably both rely on the permanent injection by expansion of an excess of magnetic energy at the largest scales, equivalent to injecting subdominant energy. Also, spectra generally steepen when initially starting flatter than k −5/3 but stop evolving at a finite time/distance.