  We define and investigate algebraic torus actions on quiver Grassmannians for
nilpotent representations of the equioriented cycle. Examples of such varieties
are type $\tt A$ flag varieties, their linear degenerations, finite dimensional
approximations of the $GL_n$-affine flag variety and affine Grassmannian. We
show that these quiver Grassmannians, equipped with our torus action, are
GKM-varieties and that their moment graph admits a combinatorial description in
terms of coefficients quiver of the underlying quiver representations. By
adapting to our setting results by Gonzales, we are able to prove that moment
graph techniques can be applied to construct module bases for the equivariant
cohomology of the above quiver Grassmannians.
