We investigate the inherently nonlinear transverse modes of the gravitational and velocity fields in ΛCDM, based on a high-resolution simulation performed using the adaptive-mesh refinement general-relativistic N-body code GRAMSES. We study the generation of vorticity in the dark matter velocity field at low redshift, providing power-law fits to the shape and evolution of its power spectrum from large sub-horizon scales to deeply nonlinear scales. By analysing the gravitomagnetic vector potential, a purely relativistic vector mode of the gravitational field that is absent in Newtonian simulations, in dark matter haloes with masses from ∼ 1012.5 h-1 M⊙ to ∼ 1015 h-1 M⊙, we find that the magnitude of this vector correlates with the halo mass, peaking in the inner regions and decreasing towards their outskirts. Nevertheless, on average, its ratio against the scalar gravitational potential remains fairly constant inside the haloes, below percent level, decreasing roughly linearly with redshift at z < 3 and showing a weak dependence on halo mass. Furthermore, we show that the gravitomagnetic acceleration in dark matter haloes peaks towards the core and reaches almost 10−10 h cm/s2 in the most massive halo of the simulation. However, regardless of the halo mass, the ratio between the magnitudes of the gravitomagneticforce and the standard gravitational force is typically at around the 10−3 level in the innermost parts of the haloes and drops by up to one order of magnitude at the outskirts. This ratio shows a very weak dependence on redshift. This result confirms that the gravitomagnetic field has a negligible effect on cosmic structure formation, even for the most massive structures, although its behaviour in low density regions such as voids remains to be explored. In addition, its effects on photons and therefore observations remains to be understood in detail in the future.