  For a smooth projective variety $X$ over a number field $k$ a conjecture of
Bloch and Beilinson predicts that the kernel of the Albanese map of $X$ is a
torsion group. In this article we consider a product $X=C_1\times\cdots\times
C_d$ of smooth projective curves and show that if the conjecture is true for
any subproduct of two curves, then it is true for $X$. Additionally, we produce
many new examples of non-isogenous elliptic curves $E_1, E_2$ with positive
rank over $\mathbb{Q}$ for which the image of the natural map
$E_1(\mathbb{Q})\otimes E_2(\mathbb{Q})\xrightarrow{\varepsilon}
\text{CH}_0(E_1\times E_2)$ is finite, including the first known examples of
rank greater than $1$. Combining the two results, we obtain infinitely many
nontrivial products $X=C_1\times\cdots\times C_d$ for which the analogous map
$\varepsilon$ has finite image.
