Let $X$ and $Y$ be regular projective varieties over $\mathbb{F}_q$ and assume we have a finite morphism $\pi: X\to Y$. Note that we can define the pushforward map on Chow group which we denote as $\pi_*$.
Now let's say $cl(C)=cl(\displaystyle\sum_i m_i \cdot C_i)$ where $cl$ is the cycle class map to $\ell$-adic etale cohomology, $C$ and $C_i$ are some cycles on $X$. Then is it true that $$cl(\pi_*(C))=cl(\displaystyle\sum_i m_i\cdot \pi_* (C_i))?$$