Unboundedness of zero-cycles on higher dimensional Fano manifolds

We show that, unlike del Pezzo surfaces, higher dimensional Fano manifolds do not satisfy in general boundedness properties for their ${\rm CH}_0$ group of $0$-cycles. For example, for quartic threefolds having a point of odd degree, there is no ``Coray type" uperbound on the minimal odd degrees of points. Also, the ${\rm CH}_0$-group of Fano hypersurfaces can be ``unbounded'' (a notion which is related to infinite dimensionality in the sense of Mumford), meaning that there is no integer $N$ such that $0$-cycles of degree at least $N$ are effective.