  For a trivial elliptic fibration $X=C \times S$ with $C$ an elliptic curve
and $S$ a projective K3 surface of Picard rank $1$, we study how various
notions of stability behave under the Fourier-Mukai autoequivalence $\Phi$ on
$D^b(X)$, where $\Phi$ is induced by the classical Fourier-Mukai
autoequivalence on $D^b(C)$. We show that, under some restrictions on Chern
classes, Gieseker semistability on coherent sheaves is preserved under $\Phi$
when the polarisation is `fiber-like'. Moreover, for more general choices of
Chern classes, Gieseker semistability under a `fiber-like' polarisation
corresponds to a notion of $\mu_\ast$-semistability defined by a `slope-like'
function $\mu_\ast$.
