In literature, the diffusion-free residence time distribution (RTD) of laminar flows – the so-called convection model – has been determined for various velocity profiles mostly on a case-by-case basis. In this analytical paper, we derive general mathematical relations which allow computing the diffusion-free differential and cumulative RTD in straight planar, circular and concentric annular channels for arbitrary monotonic and piece-wise monotonic one-dimensional velocity profiles. The theory is used to determine the RTD of plane Couette–Poiseuille flow with non-monotonic velocity profile, and the optimal value of the volumetric flow rate where the RTD becomes most narrow. It is shown that any velocity profile that depends in a sub-layer linearly on the distance from a stationary or moving no-slip wall has a differential RTD which follows a −3 power law as the residence time approaches its maximum. The variance of the RTD is directly associated with the asymptotic behavior of the RTD and can be finite or infinite.