I am wondering if, as the Quot schemes parametrizes flat (quotients of) sheaves over schemes, there is anything similar for torsion sheaves.
In first approximation, if $I$ is a sheaf of ideals over a fixed scheme $S$ (say $S$ smooth over some field $k$, for simplicity), I would like the functor $$T\mapsto \{p_2^*I-\mbox{torsion sheaves over }T\times_k S\}$$ to be representable, where $p_2$ is the second projection from $T\times_kS$ onto $S$.
Is anything of the kind true? I could be very far from the correct statement, so please feel free to propose alternatives.
Thank you in advance.