  We describe the birational correspondences, induced by the Fourier-Mukai
functor, between moduli spaces of semistable sheaves on elliptic surfaces with
sections, using the notion of $P$-stability in the derived category. We give
explicit conditions to determine whether these correspondences are
isomorphisms. This is indeed not true in general and we describe the cases
where the birational maps are Mukai flops. Moreover, this construction provides
examples of new compactifications of the moduli spaces of vector bundles via
sheaves with torsion and via complexes. We finally get for any fixed dimension
an isomorphism between the Picard groups of the moduli spaces.
