  We investigate the wall-crossing behavior as Bridgeland moduli spaces for
some Simpson moduli spaces of Gieseker-semistable torsion sheaves on
$\mathbb{P}^1\times \mathbb{P}^1$ with linear Hilbert polynomial. In
particular, we recover some of the birational transformations of these spaces
described by Maican and Chung-Moon as wall-crossing maps in certain slices of
$Stab(\mathbb{P}^1\times \mathbb{P}^1)$.
