Capillary electrophoresis (CE) and microchannel (MC) techniques are important tools in chemical and biological sciences, mainly in the study of genomes, transcriptomes, proteomes, metabolomes, and organic as well as inorganic ions. The speed of DNA sequencing increased significantly during the last decade with the use of capillary electrophoresis (CE). The time evolution of the bands' spatial profile inside capillaries and channels is of paramount importance, since the main goal of these techniques is to maximize resolution (the ratio between the spacing of the peaks and their mean standard deviations). In the present work, the propagator (retarded Green function) formalism is applied to solve a few problems which are typical for CE and MC. We also apply this mathematical method to the problem of velocity gradients along the capillary, i.e., the time evolution of the bands is analyzed when they enter regions where they migrate with different velocities.