We characterize disjoint hypercyclicity and disjoint supercyclicity of finitely many linear fractional composition operators acting on spaces of holomorphic functions on the unit disc, answering a question of Bernal. We also study mixing and disjoint mixing behaviour of projective limits of endomorphisms of a projective spectrum. In particular, we show that a linear fractional composition operator is mixing on the projective limit of the $S_v$ spaces strictly containing the Dirichlet space if and only if the operator is mixing on the Hardy space.