  We investigate a semi-continuity property for stability conditions for
sheaves that is important for the problem of variation of the moduli spaces as
the stability condition changes. We place this in the context of a notion of
stability previously considered by the authors, called
multi-Gieseker-stability, that generalises the classical notion of
Gieseker-stability to allow for several polarisations. As such we are able to
prove that on smooth threefolds certain moduli spaces of Gieseker-stable
sheaves are related by a finite number of Thaddeus-flips (that is flips arising
for Variation of Geometric Invariant Theory) whose intermediate spaces are
themselves moduli spaces of sheaves.
