We investigate the origin and the fate of high-energy photons in blazar jets, by means of first-principles particle-in-cell (PIC) kinetic simulations. In Poynting-dominated jets, magnetic reconnection is often invoked as a mechanism to dissipate the jet magnetic energy into plasma thermal energy, thus powering the observed emission. With 2D and 3D PIC simulations, we show that magnetic reconnection in blazar jets can efficiently accelerate the particles up to extreme energies, generating a flat power-law tail with slope between -2 and -1. TeV photons generated by the highest energy electrons resulting from reconnection will interact in the intergalactic medium (IGM) with the extragalactic background light, producing a dilute beam of ultra-relativistic electron-positron pairs. We study the relaxation of such beams in the IGM, by means of 2D and 3D PIC simulations. We find that at most 10% of the beam energy is ultimately transferred to the IGM plasma, so that at least 90% of the beam energy is still available to power the GeV emission produced by inverse Compton up- scattering of the CMB by the beam pairs. The energy of the TeV photons will ultimately be re- processed in the GeV band.