Contributons are the special particles distributed among a general population that generate the response observed on a specified detector. Contributon slowing-down theory describes the transfer of the response through space and energy as it is carried by contributons from the source to the detector. The response flow through space-energy and space-lethargy obeys the contributon slowing-down equation, which expresses conservation of contributons. A four-dimensional vector field is introduced to identify space and energy channels followed by the contributons, and is used to define response flow lines through space-lethargy. Numerical expressions are presented to compute the response current and slowing-down density that define the components of the response flow field. It is shown how these variables can be used to perform energy channel theory analysis of a particle transport problem. The method is applied to two realistic problems. The first determines contributon transport channels followed through space-energy by fission neutrons produced in a pressurized water reactor as they travel from the core to the reactor cavity region, where they activate surveillance dosimeters. The second examines the response transfer from a nuclear weapon burst as it is carried by contributons through space-lethargy channels in air to detectors located at some distance from ground-zero.