  We consider a set of toric Calabi-Yau varieties which arise as deformations
of the small resolutions of type A surface singularities. By careful analysis
of the heuristics of B-brane transport in the associated GLSMs, we predict the
existence of a mixed braid group action on the derived category of each
variety, and then prove that this action does indeed exist. This generalizes
the braid group action found by Seidel and Thomas for the undeformed
resolutions. We also show that the actions for different deformations are
related, in a way that is predicted by the physical heuristics.
