The modeling of steady state collisionless asymmetric tangential current layers is a challenging and poorly understood problem. For decades now, this difficulty has been limiting numerical models to approximate equilibria built with locally Maxwellian current layers and theoretical analyses to the very restricted Harris equilibrium. We show how the use of any distribution functions depending only on local macroscopic quantities results in a strong alteration of the current layer internal structure, which converges toward an unpredictable quasi-steady state with emission of ion scale perturbations. This transient can be explained in terms of ion kinetic and electron fluid physics. We demonstrate, for the first time, the validity of an asymmetric kinetic equilibrium model as well as its usability as an initial condition of hybrid kinetic simulations. This offers broad perspectives for the current sheet modeling, for which the early phase of instabilities can be studied within the kinetic formalism.