The peristaltic transport of a Casson fluid in a two-dimensional asymmetric channel is studied under long- wavelength and low-Reynolds number assumption. The asymmetry in the channel is created by considering the peristaltic waves imposed on the boundary walls to possess different amplitude and phase. The analysis of the flow is carried out in a wave frame of reference moving with the velocity of the wave. Due to the asymmetry in the channel two yield planes exist and they are calculated by solving the transcendental equation in terms of the core width. In an asymmetric channel the yield planes are skewed towards the boundary with higher amplitude or a phase difference in relation to the other boundary. While in a symmetric channel the yield planes are located symmetrically on either side of the axis of the channel. The phenomena of trapping and reflux have been discussed in the symmetric case of the channel. It is noticed that trapping of fluid occurs and the trapping zone extends for an increase in the time average flux. It is found that reflux occurs for higher values of amplitude of the peristaltic wave and the reflux zone extends for increased amplitudes.