Non-equilibrium wall-bounded turbulence is ubiquitous in geophysical and engineering flows. These include turbulence under adverse/favorable mean pressure gradients, separated flows, turbulence with mean-flow three-dimensionality, statistically unsteady turbulence, and flow over rough walls, to name a few. While many efforts have been devoted to understanding the properties and scaling of the mean velocity profiles in zero mean-pressure gradient turbulent boundary layers (ZPGTBL), non-equilibrium conditions are less explored. As such, current turbulence theories have been established mainly for ZPGTBL but not for the wide spectrum of non-equilibrium flow conditions. They expose situations where turbulence may experience counter-intuitive phenomena, such as depletion of wall shear stress, which has important implications for modeling and control. In the present work, we use a collection of building-block flows based on turbulence confined between two boundaries to analyze the scaling of the mean velocity profiles and wall shear stress in a wide range of non-equilibrium flow conditions. We leverage dimensional analysis tools to discover the most effective non-dimensional numbers and functional forms controlling the problem.