Let $X\rightarrow S$ be a holomorphic family of projective manifolds. My question is: can we find a Zariski open set $S'\subset S$ such that for any $t\in S'$, any given subvariety $Y\subset X_t$ with codimension $k$, there exists a relative subvariety $Z\subset X$ of codimension $k$ with $X_t\cap Z=Y $?
The motivation of this question is from Claire Voisin's proof of algebraic hyperbolicity of general hypersurfaces of degree large enough (in her article "On some problems of Kobayashi and Lang", proof of thm 3.3). The statement is as follows: Let $U$ be the Zariski open set parametrizing smooth hypersurfaces in $P^{n+1}$ of degree $d$. Let $\pi: X\rightarrow U, X\subset P^{n+1}\times U$ be the universal family. Assume for a general $X_t$ $(t\in U)$ there is given a subvariety $Y\subset X$ of codimension $k$. Then there exist a quasi-projective variety $B$, a dominant morphism $\phi:B\rightarrow U$, and a relative subvariety $Z\subset X_B$ of codimension $k$, where $X_B:=X\times_U B$, such that $Y\subset X_t$ is one fiber $Z_b$ for some $b\in B$.
My question is, how to prove the existence of this etale cover $\phi:B\rightarrow U$, and does this claim hold for any holomorphic family of projective manifolds?