A high-order, transient transport method based on the response matrix formalism is developed for application to reactor kinetics problems. The method combines recent advances in both static and transient response matrix methods with an explicit response-based treatment of delayed neutron precursors first proposed in the 1970s. In addition, an orthogonal basis for the time variable based on point kinetics is proposed as an alternative to a strictly polynomial basis. The method is demonstrated on infinite medium problems, the results of which show that the method can be successfully applied to reactor kinetics problems with and without precursors.