This is a question I stumbled across earlier this week. I see a similar one has been asked here.

Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\times \mathbb{A}^n_K)$ the Chow variety of effective cycles of codimension $r$ in $X\times\mathbb{A}_K^n$.

Now let's pick a closed subvariety $Z\subset X\times\mathbb{A}^n_K$ of the form $X\times D$ for $D$ an effective divisor on $\mathbb{A}^n_K$, and define

$$\text{Chow}^r(X)(Z)$$ to be the subset of $\text{Chow}^r(X)$ of the cycles whose supports intersect $Z$ in codimension higher than $k$, say ($k$ fixed). Here we assume the dimension of $X$, $n$, $r$ and $k$ are consistent.

Is $\text{Chow}^r(X)(Z)\subset \text{Chow}^r(X)$ open?

When is a restriction on the codimension of intersections with a fixed closed subvariety (e.g. codimension $\ge k$) an open condition on $\text{Chow}^r(X)$?

Rational curves on algebraic varietiesby Koll'ar) is not representable. You could just ask your question for some particular family of cycles over a specified base scheme. However, the answer is negative for the same reason as in the other comment that I wrote today. $\endgroup$also proper over the base. $\endgroup$