  We perform a study of the flavour evolution in the early universe of a
multi-flavour 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 multi-momentum
and multi-flavour 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 \sim 20 % of a degree of freedom. Such
detailed and computationally demanding treatment further raises the asymmetry
values required to significantly suppress the sterile neutrino production, up
to large and preferentially net asymmetries |L_{\nu}| > O(10^{-2}). For such
asymmetries, however, the active-sterile flavour conversions happen so late
that significant distortions are produced in the electron (anti)neutrino
spectra. The larger |L_{\nu}|, the more the impact of these distortions takes
over as dominant cosmological effect, notably increasing the 4 He abundance in
primordial nucleosynthesis (BBN). 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 possible implications for forthcoming
cosmological measurements.
