We perform a study of the flavor evolution in the early universe of a multiflavor active-sterile neutrino system with parameters inspired by the short-baseline neutrino anomalies. In a neutrino-symmetric bath a "thermal" population of the sterile state would quickly grow, but in the presence of primordial neutrino asymmetries a self-suppression as well as a resonant sterile neutrino production can take place, depending on temperature and chosen parameters. In order to characterize these effects, we go beyond the usual average momentum and single-mixing approximations and consider a multimomentum and multiflavor treatment of the kinetic equations. We find that the enhancement obtained in this case with respect to the average momentum approximation is significant, up to ~ 20% of a degree of freedom. Such a detailed and computationally demanding treatment further raises the asymmetry values required to significantly suppress the sterile neutrino production, up to L sub(nu) > ~ O(10 super(-2)). For such asymmetries, however, the active-sterile flavor conversions happen so late that significant distortions are produced in the electron (anti)neutrino spectra. The larger L sub(nu), the more the impact of these distortions takes over as a dominant cosmological effect, notably increasing the super(4)He abundance in primordial nucleosynthesis. The standard expression of the primordial yields in terms of the effective number of neutrinos and asymmetries is also greatly altered. We numerically estimate the magnitude of such effects for a few representative cases and comment on the implications for current cosmological measurements.