Let G be a reductive complex Lie group acting holomorphically on Stein manifolds X and Y. Let p_X\colon X\to Q_X and p_Y\colon Y\to 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 \phi sends the Luna strata of Q_X isomorphically onto the corresponding Luna strata of Q_Y. Fix \phi . We demonstrate two homotopy principles in this situation. The first result says that if there is a G-diffeomorphism \Phi \colon X\to Y, inducing \phi , which is G-biholomorphic on the reduced fibres of the quotient mappings, then \Phi 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 \Phi \colon X\to Y which induces a continuous family of G-equivariant biholomorphisms of the fibres p_X{^{-1}}(q) and p_Y{^{-1}}(\phi (q)) for q\in Q_X and if X satisfies an auxiliary property (which holds for most X), then \Phi is homotopic, through G-homeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y. Our results improve upon those of our earlier paper J. Reine Angew. Math. 706 (2015), 193-214 and use new ideas and techniques.