  For an abelian variety $A$ over a field $k$ the author defined in
\cite{Gazaki2015} a Bloch-Beilinson type filtration $\{F^r(A)\}_{r\geq 0}$ of
the Chow group of zero-cycles, $\text{CH}_0(A)$, with successive quotients
related to a Somekawa $K$-group. In this article we show that this filtration
behaves well with respect to isogeny, and in particular if $n:A\to A$ is the
multiplication by $n$ map on $A$, then its push-forward $n_\star$ is given on
the quotient $F^r/F^{r+1}$ by multiplication by $n^r$. In the special case when
$A=E_1\times\cdots\times E_d$ is a product of elliptic curves, we show that
this filtration agrees with a filtration defined by Raskind and Spiess and with
the Pontryagin filtration previously considered by Beauville and Bloch. We also
obtain some results in the more general case when $A$ is isogenous to a product
of elliptic curves. When $k$ is a finite extension of $\mathbb{Q}_p$, using
Jacobians of curves isogenous to products of elliptic curves, we give new
evidence for a conjecture of Raskind and Spiess and Colliot-Th\'{e}l\`{e}ne,
which predicts that the kernel of the Albanese map is the direct sum of a
divisible group and a finite group.
