The anwer is no. To see this, imagine that we have that $X = Y\times T$ (everything taken over a field, say), and $f_t$ is just the inclusion of the fibre $Y\times \{t\} \hookrightarrow Y\times T$. Take $Z$ to be some closed subscheme of $Y\times T$. We are then asking if the intersection $Z\cap (Y\times \{t\})$ is constant for an open subset of $t$.
To get a counterexample, let $Y$ be $\mathbb P^n$ (for some $n$), let $T$ be the $\mathbb P^N$ that parameterizes degree $d$ hypersurfaces in $Y$ (for some $d$), and let $Z \hookrightarrow Y\times T$ be the universal family of degree $d$ hypersurfaces. Then $Z \cap (Y\times \{t\})$ is the particular degree $d$ hypersurface corresponding to the parameter $t$, and (if $d$ is large enough) the isomorphism class of this hypersurface won't be constant on any open subset of $t$s. (One could take $n = 2$ and $d = 3$ to get a concrete example.)