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Let $k$ be an uncountable algebraically closed field of characteristic $0$,and let $X$ be a smooth projective connected variety of dimension $2p$,appropriately embedded into $\mathbb P^m$ over $k$. Let $Y$ be a hyperplanesection of $X$, and let $A^p(Y)$ and $A^{p+1}(X)$ be the groups ofalgebraically trivial algebraic cycles of codimension $p$ and $p+1$ modulorational equivalence on $Y$ and $X$ respectively. Assume that, whenever $Y$ issmooth, the group $A^p(Y)$ is regularly parametrized by an abelian variety $A$and coincides with the subgroup of degree $0$ classes in the Chow group$CH^p(Y)$. In the paper we prove that the kernel of the push-forwardhomomorphism from $A^p(Y)$ to $A^{p+1}(X)$ is the union of a countablecollection of shifts of a certain abelian subvariety $A_0$ inside $A$. For avery general section $Y$ either $A_0=0$ or $A_0$ coincides with an abeliansubvariety $A_1$ in $A$ whose tangent space is the group of vanishing cycles$H^{2p-1}(Y)_{\rm van}$. Then we apply these general results to sections of asmooth cubic fourfold in $\mathbb P^5$.

\'Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$