The authors deal with the long-time behaviour of parabolic problems with nonlinear boundary conditions. Under some natural conditions both on growth and sign conditions on nonlinearities the authors obtain uniform bounds for attractors corresponding to semigroups. Based on uniform estimates on the attractors the authors study continuity properties of the attractors for a family of problems depending on a parameter and prove upper semicontinuity of the family of attractors. Moreover the authors investigate two examples where the diffusion is perturbed: One related to homogenization theory and the other can be found, for example, in composite materials, where the heat diffusion properties can change significantly from one part of the domain to another.