Dislocation climb is an important high temperature process of metals plasticity, responsible for phenomena such as creep, swelling, or hardening. Climb is defined by the ability of dislocations to leave their original glide plane by interaction with point defects. As such, dislocation climb is controlled by point defect diffusion/absorption/emission, which are thermally-activated processes. While thermodynamically-consistent models for climb have been developed, they are formulated in a continuum framework, defining effective defect fluxes and climb propensities in response to thermodynamic driving forces. However, the point-wise nature of vacancies (and/or self-interstitials) confers a highly discrete nature to climb dynamics, which is also strongly affected by elastic forces. The combination of discreteness, thermal activation, and elasticity is too much to handle for direct atomistic methods such as molecular dynamics, or elasticity methods such as dislocation dynamics. Here we develop a kinetic Monte Carlo module of vacancy generation and transport kinetics acting on top of evolving elastic fields provided by discrete dislocation dynamics simulations. The coupling between both is defined by the applied stresses and the stress gradients generated by dislocation structures. We employ the method to study elementary climb processes in iron crystals and furnish climb mobility functions to be used in parametric dislocation dynamics simulations. We apply the technique to study nonconservative plastic bypass of spherical precipitates by edge dislocations and point out the differences between our discrete approach and existing continuum formulations.