  We study generic semilinear Schr\"odinger systems which may be written in
Hamiltonian form. In the presence of a single gauge invariance, the components
of a solution may exchange mass between them while preserving the total mass.
We exploit this feature to unravel new orbital instability results for
ground-states. More precisely, we first derive a general instability criterion
and then apply it to some well-known models arising in several physical
contexts. In particular, this mass-transfer instability allows us to exhibit
$L^2$-subcritical unstable ground-states.
