We have investigated magnetic field generation in velocity shears via the kinetic Kelvin–Helmholtz instability (kKHI) using a relativistic plasma jet core and stationary plasma sheath. Our three-dimensional particle-in-cell simulations consider plasma jet cores with Lorentz factors of 1.5, 5, and 15 for both electron–proton andelectron–positron plasmas. For electron–proton plasmas, we find generation of strong large-scale DC currents and magnetic fields that extend over the entire shear surface and reach thicknesses of a few tens of electron skin depths. For electron–positron plasmas, we find generation of alternating currents and magnetic fields. Jet and sheath plasmas are accelerated across the shear surface in the strong magnetic fields generated by the kKHI. The mixing of jet and sheath plasmas generates a transverse structure similar to that produced by the Weibel instability.