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Abstract Astrophysical relativistic outflows are launched as Poynting-flux dominated, yet the mechanism governing efficient magnetic dissipation, which powers the observed emission, is still poorly understood. We study magnetic energy dissipation in relativistic “striped” jets, which host current sheets separating magnetically dominated regions with opposite field polarity. The effective gravity forcegin the rest frame of accelerating jets drives the Kruskal–Schwarzschild instability (KSI), a magnetic analog of the Rayleigh–Taylor instability. By means of 2D and 3D particle-in-cell simulations, we study the linear and nonlinear evolution of the KSI. The linear stage is well described by linear stability analysis. The nonlinear stages of the KSI generate thin (skin-depth-thick) current layers, with length comparable to the dominant KSI wavelength. There, the relativistic drift-kink mode and the tearing mode drive efficient magnetic dissipation. The dissipation rate can be cast as an increase in the effective width Δ_effof the dissipative region, which follows $d{\Delta }_{\mathrm{eff}}/dt\simeq 0.05\sqrt{{\Delta }_{\mathrm{eff}}g}$. Our results have important implications for the location of the dissipation region in gamma-ray burst and active galactic nuclei jets.