Abstract. Let G be a reductive complex Lie group acting holomorphically on Stein manifolds X and Y . Let p X : X → Q X and p Y : Y → Q Y be the quotient mappings. When is there an equivariant biholomorphism of X and Y ? A necessary condition is that the categorical quotients Q X and Q Y are biholomorphic and that the biholomorphism ϕ sends the Luna strata of Q X isomorphically onto the corresponding Luna strata of Q Y . Fix ϕ. We demonstrate two homotopy principles in this situation. The first result says that if there is a G-diffeomorphism Φ : X → Y , inducing ϕ, which is G-biholomorphic on the reduced fibres of the quotient mappings, then Φ is homotopic, through G-diffeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y . The second result roughly says that if we have a G-homeomorphism Φ : X → Y which induces a continuous family of G-equivariant biholomorphisms of the fibres p X −1 (q) and p Y −1 (ϕ(q)) for q ∈ Q X and if X satisfies an auxiliary property (which holds for most X), then Φ is homotopic, through Ghomeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y . Our results improve upon those of [KLS15] and use new ideas and techniques.