Many Navier-Stokes Cahn-Hilliard (NSCH) models with non-matching densities have been proposed over the last decades, see e.g. [1,2]. Even though these models aim to represent the same physical phenomena, they seem to differ at first sight. The first aim of this talk is to present a framework for NSCH models that unites all of these models [3]. Our development is based on three unifying principles: (1) there is only one system of balance laws based on continuum mixture theory that describes the physical model, (2) there is only one natural energy-dissipation law that leads to quasi-incompressible NSCH models, (3) variations between the models only appear in the constitutive choices. Our framework for NSCH models is not fully compatible with mixture theory. Namely, it represents reduced models in which the evolution equations for the diffusive fluxes are replaced by constitutive models. The second aim of this talk is to present a new incompressible model that guarantees compatibility with mixture theory by replacing the energy-dissipation law by the second law of thermodynamics [4]. We compare this model, analytically and computationally, to existing NSCH models. We conclude by discussing a divergence-conforming isogeometric finite element discretization, and showing relevant benchmark computations such as a rising air bubble in water [5]. [1] Lowengrub, J.S., Truskinovsky, L., Quasi–incompressible Cahn–Hilliard fluids and topological transitions, Proceedings of the Royal Society of London. Series A: Math- ematical, Physical and Engineering Sciences, 454 (1998), 2617–2654. [2] Aki, G.L. Dreyer, W., Giesselmann J. and Kraus C., A quasi-incompressible diffuse interface model with phase transition, Mathematical Models and Methods in Applied Sciences 24 (2014), 827–861. [3] ten Eikelder, M.F.P., van der Zee, K.G., Akkerman, I. and Schillinger, D., A unified framework for Navier-Stokes Cahn-Hilliard models with non-matching densities. Mathematical Models and Methods in Applied Sciences 33 (2023), 175-221. [4] ten Eikelder, M.F.P., van der Zee, K.G., and Schillinger, D., Thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids. arXiv preprint arXiv:2302.09287. [5] ten Eikelder, M.F.P. and Schillinger, D., The divergence-free velocity formulation of the consistent Navier-Stokes Cahn-Hilliard model with non-matching densities, divergence-conforming discretization, and benchmarks. arXiv preprint arXiv:2311.09966.