Tight closure/localization problem/geometric deformation/short description

Let ${}\mathbb {Z} /(p)\subset D$ (think of ${}\mathbb {Z} /(p)[t]$) and let ${}D\subseteq R$ be flat, ${}f$ and ${}I$ in ${}R$.

For every ${}D\rightarrow K$, ${}K$ a field,

we can consider ${}f$ and ${}I$ in ${}R\otimes _{D}K$.

In particular for ${}K=\kappa ({\mathfrak {p}})$, ${}{\mathfrak {p}}\in \operatorname {Spec} \,D$.

How does the property

$f\in I^{*}{\text{ in }}R\otimes _{D}\kappa ({\mathfrak {p}})\,$

vary with ${}{\mathfrak {p}}$?

We call such a situation a geometric or equicharacteristic deformation of tight closure problems.