A fraction of the active supermassive black holes at the centers of galaxies in our Universe are capable of launching extreme kiloparsec-long relativistic jets. These jets are known multiband (radio to γ-ray) and multimessenger (neutrino) emitters, and some of them have been monitored over decades at all accessible wavelengths. However, several open questions remain unanswered about the processes powering these highly energetic phenomena. These jets intrinsically produce soft-to-hard X-ray emission that extends from E>0.1keV up to E>100keV, and simultaneous broadband X-ray coverage, combined with excellent timing and imaging capabilities, is required to uncover the physics of jets. Indeed, truly simultaneous soft-to-hard X-ray coverage, in synergy with current and upcoming high-energy facilities (such as IXPE, COSI, CTAO, etc.) and neutrino detectors (e.g., IceCube), would enable us to disentangle the particle population responsible for the high-energy radiation from these jets. A sensitive hard X-ray survey (F_20-80keV<10^-15erg cm^-2s^-1) could unveil the bulk of their population in the early Universe. Acceleration and radiative processes responsible for the majority of their X-ray emission would be pinned down by microsecond timing capabilities at both soft and hard X-rays. Furthermore, imaging jet structures for the first time in the hard X-ray regime could unravel the origin of their high-energy emission. The proposed Probe-class mission concept High Energy X-ray Probe (HEX-P) combines all these required capabilities, making it the crucial next-generation X-ray telescope in the multi-messenger, time-domain era. HEX-P will be the ideal mission to unravel the science behind the most powerful accelerators in the Universe.