  Generalized affine Grassmannian slices provide geometric realizations for
weight spaces of representations of semisimple Lie algebras. They are also
Coulomb branches, symplectic dual to Nakajima quiver varieties. In this paper,
we prove that neighbouring generalized affine Grassmannian slices are related
by Hamiltonian reduction by the action of the additive group. We also prove a
weaker version of the same result for their quantizations, algebras known as
truncated shifted Yangians.
