Let Ω₀ be a bounded domain in ℝⁿ, let G be a family of diffeomorphisms, and let Ω_G = G(Ω ₀) for G ∈ G. Denote by E^t(G) the semigroup generated by a fixed parabolic PDE with Dirichlet boundary conditions on the boundary of Ω_G. Let A_G be the global attractor of E _t(G). Conditions are given under which a generic diffeomorphism G ∈ G is a continuity point of the map G → A_G.