We numerically investigate the long-term temporal evolution of magnetized jets where the computational domain covers multiple wavelengths (up to 10) of the fastest growing Kelvin-Helmholtz unstable mode. The dynamical importance of the magnetic field, which is initially uniform and flow-aligned, varies over a significant range: the plasma beta in the jets ranges from O(1000) (essentially hydrodynamical) down to O(1) (equipartition jets). Our calculations of two-dimensional, longitudinally periodic, extended slab configurations identify an inverse cascade process in the overall disruption to a broadened and heated jet flow. This process occurs for transonic and supersonic flows as well, with rapid shock-dominated transients appearing in supersonic cases, and with characteristic differences depending on the initial jet width. For configurations with a jet velocity profile having a radius that is much larger than the vorticity thickness of the flow, the cascade proceeds early through pairing/merging of individual mode structures on both jet boundaries. Jets with radii of the order of the vorticity thickness are strongly unstable to sinuous deformations with boundary layer-layer interactions between vortex (transonic, weak magnetic field) and shock (supersonic, strong field) structures in a few sound crossing times. We back up these findings for planar jets with selected three-dimensional simulations of extended cylindrical jet configurations. These tend to have more small-scale fluctuations in their relaxed endstates. The timescales and overall scenario for the helical disruptions agree well with the 2D studies. This allows us to discuss the possible implications of our results in the context of magnetohydrodynamic stability of astrophysical jets.