We present an approximation for the semigroup generated by a multiplicative perturbation of a Dirichlet-to-Neumann operator and more general Poincaré–Steklov operators which are in a certain relation to a generator of a semigroup of composition operators. This approach gives in particular an approximation of the semigroup generated by the classical Dirichlet-to-Neumann operator on a Dini-smooth Jordan domain in terms of composition operators.