The variation of the Gysin kernel in a family

Consider a smooth projective surface $S$. Consider a fibration $S\to C$ where $C$ is a quasi-projective curve such the fibers are smooth projective curves. The aim of this text is to show that the kernels of the push-forward homomorphism $\{j_{t*}\}_{t\in C}$ from the Jacobian $J(C_t)$ to $A_0(S)$ forms a family in the sense that it is a countable union of translates of an abelian scheme over $C$ sitting inside the Jacobian scheme $\mathscr{J}\to C$, such that the fiber of this countable union at $t$ is the kernel of $j_{t*}$.