Suppose $f:X \rightarrow Y$ is morphism between non-singular projective varieties, and $Z$ is a codimension-$c$ cycle of $Y$. From the chapter on Intersection Theory of Stack Project, for rational equivalence relation, the pull back of the cycle class $[Z]$ represented by $Z$ is \begin{equation} f^*([Z])=(pr_X)_*([\Gamma_f] \cdot (pr_Y)^*[Z]) \end{equation} When the scheme theoretic inverse image $f^{-1}(Z)$ is also of codimension $c$ and $Z$ is Cohen-Macaulay at the images of the generic points of $f^{-1}(Z)$, then the pull-back $f^*[Z]$ is just the cycle associated to the scheme $[f^{-1}(Z)]$.
Question: for an arbitrary cycle $Z$, does there exist a cycle $Z'$ which is rationally equivalent to $Z$ such that the codimension of $f^{-1}(Z')$ is $c$ and \begin{equation} f^*([Z])=[f^{-1}(Z')] \end{equation}