  We study the twisted index of 3d $\mathcal{N}=2$ supersymmetric gauge
theories on $S^1 \times \Sigma$ in the presence of a real FI parameter
deformation. This parameter induces a 1d FI parameter for the effective
supersymmetric quantum mechanics on $S^1$. Using supersymmetric localisation,
the twisted index can be expressed as a contour integral. We show that the
contour prescription is modified in the presence of the 1d FI parameter,
leading to wall-crossing phenomena for the twisted index. In particular, we
derive a general wall-crossing formula for abelian gauge theories. We also
examine the origin of wall-crossing as change of stability condition in the
algebro-geometric interpretation of the twisted index. These ideas are
illustrated for abelian theories with $\mathcal{N}=4$ supersymmetry and in a
non-abelian example that reproduces wall-crossing phenomena associated to
moduli spaces of stable pairs.
