We consider the discrete, time-varying broadcast channel with memory under the assumption that the channel states belong to a set of finite cardinality. We study the achievable rates in two scenarios where feedback (and cooperation) is available. One scenario is the general finite-state broadcast channel (FSBC) where both receivers send feedback to the transmitter, and in addition one receiver sends his channel outputs to the other receiver through a cooperation link. The second scenario is the degraded FSBC where only the strong receiver sends feedback to the transmitter. We find the capacity regions for both cases. In both scenarios we consider non-indecomposable as well as a class of indecomposable FSBCs.