Let $f: X \to Y$ be a proper morphism of smooth integral projective varieties over an algebraically closed field $k$ of characteristic 0, where $dim X = dim Y = n$. Denote by $CH_i(W):= Z_i(W)/\sim$ be the Chow group of $i$-cycles on $W$, where $Z^i(W)$ is the group of $i$-cycles on $W$ and $\sim$ means rational equivalence. The push-forward $f_\ast: Z_i(X) \to Z_i(Y)$ is a group homomorphism that is compatible with rational equivalence, meaning that it descends to a group homomorphism $$ f_\ast : CH_i(X) \to CH_i(Y).$$ When $i = n-1$, under the above assumptions on $X$ and $Y$ we have that $CH_{n-1}(X) =CaCl(X) = Pic(X)$ and $CH_{n-1}(Y) =CaCl(Y)=Pic(Y)$, since rational equivalence is the same as linear equivalence. Hence, $f_\ast$ is a group homomorphism $$ f_\ast : Pic(X) \to Pic(Y).$$
Assuming that $Pic(Y)$ and $Pic(X)$ are representable by algebraic group schemes (which they are, under our assumptions), is $ f_\ast : Pic(X) \to Pic(Y)$ a morphism also as algebraic group schemes?
Moreover, I'm slightly confused about one thing: usually the pushforward of a line bundle is not a line bundle, meaning that the above morphism $ f_\ast : Pic(X) \to Pic(Y)$ "looks" wrong! Hence,
is the word "pushforward" an abuse of terminology in the context of cycles? i.e. do pushforwards of invertible line bundles actually correspond to pushforwards of cycles? If not, is there any relation between the two?