  In this paper general dynamic equations describing the time evolution of
isothermal quasi-incompressible multicomponent liquids are derived in the
framework of the classical Ginzburg-Landau theory of first order phase
transformations. Based on the fundamental {equations of continuum mechanics}, a
general convection-diffusion dynamics is set up first for compressible liquids.
The constitutive relations for the diffusion fluxes and the capillary stress
are determined in the framework of gradient theories. {Next the general
definition of incompressibility is given}, which is taken into account {in the
derivation} by using the Lagrange multiplier method. To validate the theory,
the dynamic equations are solved numerically for the quaternary
quasi-incompressible Cahn-Hilliard system. It is demonstrated that variable
density (i) has no effect on equilibrium (in case of a suitably constructed
free energy functional), {and (ii) can} influence non-equilibrium pattern
formation significantly.
