Hyperbolic balance laws can have particular moving equilibria involving particular source terms or based on divergence-free structures. Maintaining at a discrete level such equilibria (either exactly or very accurately) can give strong advantages with respect to classical methods. The Global Flux technique has been introduced to preserve 1D equilibria with source integrating the source into the flux term and has shown an increased accuracy in combination with many discretizations (FV, FD, FEM, DG). The 2D extensions have shown promising results in the linear acoustics case for vortex-like solutions. We will see what are the limits extending this method to nonlinear and multidimensional problems.