Non-relativistic, magnetically-driven jets are constructed by taking self-consistently into account the feedback on the underlying accretion disc. It is shown that such jets are mostly described by the ejection index $��= \mbox{d} \ln \dot M_a/ \mbox{d}\ln r$, which is a local measure of the disc ejection efficiency. This parameter is found to lie in a very narrow range, due to global disc-jets constraints. This investigation provided two important results. First, the disc vertical equilibrium imposes a minimum mass flux ejected. Second, their asymptotic behaviour critically depends on a fastness parameter $��_A= ��_*r_A /V_{Ap,A}$, ratio of the field lines rotation velocity to the poloidal Alfv��n velocity at the Alfv��n surface. This parameter must be bigger than, but of the order of, unity. Self-similar jets from Keplerian discs, after widening up to a maximum radius whose value increases with $��_A$, always recollimate towards the jet axis, until the fast-magnetosonic critical point is reached. It is doubtful that such solutions could steadily cross this last point, the jet either ending there or rebouncing. Recollimation takes place because of the increasing effect of magnetic constriction. This systematic behaviour is due to the large opening of the magnetic surfaces, leading to such an efficient acceleration that matter always reaches its maximum poloidal velocity. This ``over-widening'' stems from having the same ejection efficiency $��$ in the whole jet. Realistic jets, fed with ejection indices varying from one magnetic surface to the other, would not undergo recollimation, allowing either cylindrical or parabolic asymptotic collimation.

21 pages, 13 Postscript figures, accepted in A&A