We consider a model of non-isothermal phase separationtaking place in a confined container. The order parameter is governed by a viscous or nonviscous Cahn-Hilliard type equation which is coupled with a heat equation for the temperature . The former is subject to a nonlinear dynamic boundary condition recently proposed by physicists to account for interactions with the walls, while the latter is endowed with a standard (Dirichlet, Neumann or Robin) boundary condition. We analyze issues like well-posedness and the asymptotic behavior of the solutions within the theory of infinite-dimensional dynamical systems (that is, global and exponential attractors and their stability with respect to some physical parameters). We also intend to present results about convergence to equilibria.