  One view of the solar-wind turbulence is that the observed highly anisotropic
fluctuations at spatial scales near the proton inertial length $d_p$ may be
considered as Kinetic Alfv\'en waves (KAWs). In the present paper, we show how
phase-mixing of large-scale parallel propagating Alfv\'en waves is an efficient
mechanism for the production of KAWs at wavelengths close to $d_p$ and at large
propagation angle with respect to the magnetic field. Magnetohydrodynamic
(MHD), Hall-Magnetohydrodynamic (HMHD), and hybrid Vlasov-Maxwell (HVM)
simulations modeling the propagation of Alfv\'en waves in inhomogeneous plasmas
are performed. In linear regime, the role of dispersive effects is singled out
by comparing MHD and HMHD results. Fluctuations produced by phase-mixing are
identified as KAWs through a comparison of polarization of magnetic
fluctuations and wave group velocity with analytical linear predictions. In the
nonlinear regime, comparison of HMHD and HVM simulations allows to point out
the role of kinetic effects in shaping the proton distribution function. We
observe generation of temperature anisotropy with respect to the local magnetic
field and production of field-aligned beams. The regions where the proton
distribution function highly departs from thermal equilibrium are located
inside the shear layers, where the KAWs are excited, this suggesting that the
distortions of the proton distribution are driven by a resonant interaction of
protons with KAW fluctuations. Our results are relevant in configurations where
magnetic field inhomogeneities are present, as, for example, in the solar
corona where the presence of Alfv\'en waves has been ascertained.

Solar and Stellar Astrophysics;Plasma Physics