Based on the method of moments, we derive a general theoretical expression for the time-dependent dispersion of an initial point concentration in steady and unsteady laminar flows through long straight channels of any constant cross-section. We retrieve and generalize previous case-specific theoretical results, and furthermore predict new phenomena. In particular, for the transient phase before the well-described steady Taylor–Aris limit is reached, we find anomalous diffusion with a dependence of the temporal scaling exponent on the initial release point, generalizing this finding in specific cases. During this transient we furthermore identify maxima in the values of the dispersion coefficient which exceed the Taylor–Aris value by amounts that depend on channel geometry, initial point release position, velocity profile and Péclet number. We show that these effects are caused by a difference in relaxation time of the first and second moments of the solute distribution and may be explained by advection-dominated dispersion powered by transverse diffusion in flows with local velocity gradients.