We study Langevin dynamics with a kinetic energy different from the standard, quadratic one in order to accelerate the sampling of Boltzmann-Gibbs distributions. In particular, this kinetic energy can be nonglobally Lipschitz, which raises issues for the stability of discretizations of the associated Langevin dynamics. We first prove the exponential convergence of the law of the continuous process to the Boltzmann-Gibbs measure by a hypocoercive approach and characterize the asymptotic variance of empirical averages over trajectories. We next develop numerical schemes which are stable and of weak order two by considering splitting strategies where the discretizations of the fluctuation/dissipation are corrected by a Metropolis procedure. We use the newly developed schemes for two applications: optimizing the shape of the kinetic energy for the so-called adaptively restrained Langevin dynamics (which considers perturbations of standard quadratic kinetic energies vanishing around the origin) and reducing the metastability of some toy models using nonglobally Lipschitz kinetic energies.