Let $X$ and $Y$ be complex projective varieties. Let's assume we have a finite flat morphism $f:X\rightarrow Y$ of degree $k$. We know that it is possible to pullback and also pushforward algebraic cycles along $f$. These pullback and pushforward of cycles also induces morphisms on Chow groups. I was wondering whether this also induces regular morphisms on Chow varieties or not. More precisely let $C_{r,d}(Y)$ and $C_{r,d}(X)$ be the Chow variety of degree $d$ and $r$ dimensional cycles on $X$ and $Y$ respectively. Note that degree of cycles depends on the projective embedding.

1. Does pullback induce a regular morphism of Chow varieties from $C_{r,d}(Y)$ to $C_{r, kd'}(X)$? Note that $d'$ is not necessarily same as $d$ since the degrees can change depending on the embedding, I am tempted to assume that there is some embedding for the pair $X$ and $Y$ such that one can assume $d=d'$ but I decided to not to assume that.

2. Does pushforward induce a regular morphism of Chow varieties like $C_{r, d}(X) \rightarrow C_{r, d'}(Y)$? again the $d'$ can be something other than $d$ depending on the embedding of varieties.

3. If the answer to (2) is positive is the map also a flat morphism?

Let $X$ and $Y$ be complex projective varieties. Let's assume we have a finite flat morphism $f:X\rightarrow Y$ of degree $k$. We know that it is possible to pullback and also pushforward algebraic cycles along $f$. These pullback and pushforward of cycles also induces morphisms on Chow groups. I was wondering whether this also induces regular morphisms on Chow varieties or not. More precisely let $C_{r,d}(Y)$ and $C_{r,d}(X)$ be the Chow variety of degree $d$ and $r$ dimensional cycles on $X$ and $Y$ respectively. Note that degree of cycles depends on the projective embedding.

The two questions below were answered positively:

1) Does pullback induce a regular morphism of Chow varieties from $C_{r,d}(Y)$ to $C_{r, kd'}(X)$? Note that $d'$ is not necessarily same as $d$ since the degrees can change depending on the embedding, I am tempted to assume that there is some embedding for the pair $X$ and $Y$ such that one can assume $d=d'$ but I decided to not to assume that.

2) Does pushforward induce a regular morphism of Chow varieties like $C_{r, d}(X) \rightarrow C_{r, d'}(Y)$? again the $d'$ can be something other than $d$ depending on the embedding of varieties.

Let's consider the push-forward map $f_*:C_{r, d}(X) \rightarrow C_{r, d'}(Y)$. Is $f_*$ a flat morphism? Furthermore you can assume $f$ is a quotient map under some finite group action. If not, does it become flat after restricting to the Chow variety of irreducible cycles?