  We describe the Griffiths group of the product of a curve $C$ and a surface
$S$ as a quotient of the Albanese kernel of $S$ over the function field of $C$.
When $C$ is a hyperplane section of $S$ varying in a Lefschetz pencil, we prove
the nonvanishing in $\text{Griff}(C\times S)$ of a modification of the graph of
the embedding $C\hookrightarrow S$ for infinitely many members of the pencil,
provided the ground field $k$ is of characteristic $0$, the geometric genus of
$S$ is $>0$, and $k$ is large or $S$ is "of motivated abelian type".
