• 1 Instituto de Matemática da Universidade Federal do Rio de Janeiro
• 2 Instituto de Matemática, Estatística e Computação Científica [Brésil]
• 3 Laboratoire de Mathématiques de Bretagne Atlantique [LMBA]

Any toric flip naturally induces an equivalence between the associated categories of equivariant reflexive sheaves, and we investigate how slope stability behaves through this functor. On one hand, for a fixed toric sheaf, and natural polarisations that make the exceptional loci small, we provide a simple numerical criterion that characterizes when slope stability is preserved through the flip. On the other hand, for a given flip, we introduce full subcategories of logarithmic toric sheaves and characterize when polystability is preserved for all toric sheaves in those subcategories at once.