  In this paper we characterize mixing composition operators acting on the
space $\mathscr{O}_M(\mathbb{R})$ of slowly increasing smooth functions.
Moreover we relate the mixing property of those operators with the solvability
of Abel's functional equation and we give a sufficient condition for sequential
hypercyclicity of composition operators on $\mathscr{O}_M(\mathbb{R})$. This is
used to prove that many mixing composition operators are hypercyclic.
