Let $B$ be a smooth quasi-projective variety over a field of characteristic zero. Let $\pi\colon \mathcal{X} \rightarrow B$ be a smooth and projective morphism with geometrically integral fibres. Let $Z$ be a cycle on $\mathcal{X}$, i.e. a $\mathbb{Z}$-linear combination of closed integral subvarieties of $\mathcal{X}$. Suppose in addition that $Z$ is flat over $B$, i.e. each component of the support of $Z$ is flat over $B$.
Question: is the locus $B_{\text{trivial}} = \{ b\in B \mid \text{ the cycle }Z_b \text{ of }\mathcal{X}_b \text{ is rationally trivial}\}$ a closed subset of $B$?
(Recall that a cycle on a variety $X$ is called rationally trivial if it lies in the subgroup of cycles of the form $W_0-W_\infty$ where $W$ is a cycle $X\times \mathbb{P}^1$ flat over $\mathbb{P}^1$.)
I can show that $B_{\text{trivial}}$ is closed under specialization, since by smoothness it suffices to consider the case of a DVR and in that case there is a well-defined specialization morphism on the level of Chow groups, see the last chapter of Fulton's intersection theory book. So it suffices to prove that $B_{\text{trivial}}$ is a constructible set. Any reference/help would be appreciated!
