Skin-friction decompositions such as the so-called FIK identity (Fukagata et al., 2002) are useful tools in identifying relevant contributions to the friction, but may also lead to results difficult to interpret when the total friction is recovered from cancellation of multiple terms with large values. We propose a new formulation of the FIK contributions related to streamwise inhomogeneity, which is derived from the convective form of the momentum equation and using the concept of dynamic pressure. We examine turbulent boundary layers subjected to various pressure-gradient conditions, including cases with drag-reducing control. The new formulation distinguishes more precisely the roles of the free-stream pressure distribution, wall-normal convection, and turbulent fluctuations. Our results allow to identify different regimes in adverse-pressure-gradient turbulent boundary layers, corresponding to different proportions of the various contributions, and suggest a possible direction towards studying the onset of mean separation.