When $X$ is a Grassmannian, Marian-Oprea-Pandharipande and Toda constructed alternate compactifications of spaces of maps from curves to $X$. The construction has been generalized to a large class of GIT quotients $X=W//G$ by Ciocan-Fontanine-Kim-Maulik and many others. It is called the theory of $\epsilon$-stable quasimaps. In this talk, we will introduce permutation-equivariant K-theoretic epsilon-stable quasimap invariants and prove their wall-crossing formulae for all targets in all genera. The wall-crossing formulae generalize Givental's K-theoretic toric mirror theorem in genus zero. In physics literature, these K-theoretic invariants are related to the $3d N = 2$ supersymmetric gauge theories studied by Jockers-Mayr, and the wall-crossing formulae can be interpreted as relations between invariants in the UV and the IR phases of the $3d$ gauge theory. It is based on joint work with Yang Zhou.