Let $f : X\rightarrow S$ be a flat finite type morphism of schemes with $S$ integral and Noetherian. Let $\eta\in S$ be the generic point.

Let $\{\sigma_i\}$ be a collection of sections of $f$ (possibly infinite), which are Zariski dense in $X_\eta$. I'm interested in additional conditions on $f,\{\sigma_i\}$ under which one or both of the following properties are satisfied:

**P1: There exists a point $s\in S - \{\eta\}$ such that $\{\sigma_i\}$ is dense in the fiber $X_s$.**

**P2: There exists a nonempty open $U\subset S$ such that P1 holds for every $s\in U$.**

Clearly a necessary condition is that $S - \{\eta\}$ needs to be ``large'' (e.g., P1 will often fail if $S$ is the spec of a discrete valuation ring). From now on lets assume $S - \{\eta\}$ is Zariski dense in $S$.

My intuition is that under mild additional assumptions there should be some kind of semicontinuity result for the dimensions of the Zariski closures inside fibers. In particular, the set of $s\in S$ such that $\{\sigma_i\}$ is not Zariski dense in $X_s$ should be closed in $S$. However I'm not aware of any results in this direction.

Here are some specific questions:

(1) Do P1, P2 hold under the above assumptions?

(2) What if we also assume $f$ has geometrically irreducible fibers?

(3) What if $X$ is a semisimple affine algebraic $S$-group scheme?