Buoyancy-driven exchange flows arise in the natural and built environment wherever bodies of fluids at different densities are connected by a narrow constriction. In this paper we study these flows in the laboratory using the canonical stratified inclined duct experiment, which sustains an exchange flow in an inclined duct of rectangular cross-section over long time periods (Meyer & Linden, J. Fluid Mech., vol. 753, 2014, pp. 242–253). We study the behaviour of these sustained stratified shear flows by focusing on three dependent variables of particular interest: the qualitative flow regime (laminar, wavy, intermittently turbulent or fully turbulent), the mass flux (net transport of buoyancy between reservoirs) and the interfacial thickness (thickness of the layer of intermediate density between the two counter-flowing layers). Dimensional analysis reveals five non-dimensional independent input parameters: the duct aspect ratios in the longitudinal direction
$A$
and spanwise direction
$B$
, the tilt angle
$\unicode[STIX]{x1D703}$
, the Reynolds number
$Re$
(based on the initial buoyancy difference driving the flow) and the Prandtl number
$Pr$
(we consider both salt and temperature stratifications). After reviewing the literature and open questions on the scaling of regimes, mass flux and interfacial thickness with
$A,B,\unicode[STIX]{x1D703},Re,Pr$
, we present the first extensive, unified set of experimental data where we varied systematically all five input parameters and measured all three output variables with the same methodology. Our results in the
$(\unicode[STIX]{x1D703},Re)$
plane for five sets of
$(A,B,Pr)$
reveal a variety of scaling laws, and a non-trivial dependence of all three variables on all five parameters, in addition to a sixth elusive parameter. We further develop three classes of candidate models to explain the observed scaling laws: (i) the recent volume-averaged energetics of Lefauve et al. (J. Fluid Mech., vol. 848, 2019, pp. 508–544); (ii) two-layer frictional hydraulics; (iii) turbulent mixing models. While these models provide significant qualitative and quantitative descriptions of the experimental results, they also highlight the need for further progress on shear-driven turbulent flows and their interfacial waves, layering, intermittency and mixing properties.