  We consider elliptic fibrations with arbitrary base dimensions, and
generalise previous work by the second author. In particular, we check
universal closedness for the moduli of semistable objects with respect to a
polynomial stability that reduces to PT-stability on threefolds. We also show
openness of this polynomial stability. On the other hand, we write down
criteria under which certain 2-term polynomial semistable complexes are mapped
to torsion-free semistable sheaves under a Fourier-Mukai transform. As an
application, we construct an open immersion from a moduli of complexes to a
moduli of Gieseker stable sheaves on higher dimensional elliptic fibrations.
