Surfactant transport is central to a diverse range of natural phenomena with numerous practical applications in physics and engineering. Surprisingly, this process remains relatively poorly understood at the molecular scale. Here, we use non-equilibrium molecular dynamics (NEMD) simulations to study the spreading of sodium dodecyl sulphate on a thin film of liquid water. The molecular form of the control volume is extended to a coordinate system moving with the liquid–vapour interface to track surfactant spreading. We use this to compare the NEMD results to the continuum description of surfactant transport on an interface. By including the molecular details in the continuum model, we establish that the transport equation preserves substantial accuracy in capturing the underlying physics. Moreover, the relative importance of the different mechanisms involved in the transport process is identified. Consequently, we derive a novel exact molecular equation for surfactant transport along a deforming surface. Close agreement between the two conceptually different approaches, i.e. NEMD simulations and the numerical solution of the continuum equation, is found as measured by the surfactant concentration profiles, and the time dependence of the so-called spreading length. The current study focuses on a relatively simple specific solvent–surfactant system, and the observed agreement with the continuum model may not arise for more complicated industrially relevant surfactants and anti-foaming agents. In such cases, the continuum approach may fail to predict accompanying phase transitions, which can still be captured through the NEMD framework.