I have heard several times that the proof of the second part of the Chow's moving lemma (of algebraic geometry), is problematic; and that this is the reason Fulton, Intersection theory, does not use it to define intersection product.

Has anyone heard similar claims, and if so, what/where is the gap ?

For completeness, I mean the moving lemma, say as stated in J.Roberts, Appendix 2(Algebraic Geometry, 1970, ed. Oort). The problem claimed is with the 'Moreover' part.

Lemma (Chow's moving lemma). - Let $W$ and $W'$ be cycles on X. Then there exists a cycle $W''$ rationally equivalent to $W'$ such that $W \cdot W''$ is defined. Moreover, there exists an algebraic family $\{Z(t)\}_{t \in P^1(k)}$ deforming $W'$ into $W$ such that $Z(t)\cdot W$ is defined for almost all t.