In this paper, we study quasicompact composition endomorphisms of Lipschitz algebras of complex-valued bounded functions on metric spaces, not necessarily compact. We give some sufficient conditions that a composition endomorphism of these algebras to be quasicompact. We also establish an upper bound and a formula for the essential spectral radius of a composition endomorphism $ T $ of these algebras under some conditions which implies that $ T $ is quasicompact. d
