  For an equivariant reflexive sheaf over a polarised toric variety, we study
slope stability of its reflexive pullback along a toric fibration. Examples of
such fibrations include equivariant blow-ups and toric locally trivial
fibrations. We show that stability (resp. unstability) is preserved under such
pullbacks for so-called adiabatic polarisations. In the strictly semistable
situation, under locally freeness assumptions, we provide a necessary and
sufficient condition on the graded object to ensure stability of the pulled
back sheaf. As applications, we provide various stable perturbations of
semistable tangent sheaves, either by changing the polarisation, or by
blowing-up a subvariety. Finally, our results apply uniformly in specific flat
families and induce injective maps between the associated moduli spaces.
