Light dark matter (LDM) in the mega-electron-volt to giga-electron-volt mass region is an attractive candidate for the all-pervasive and encompassing matter making up the vast bulk of the mass of our universe. Beam dump experiments at high-intensity accelerators are a powerful tool to produce and detect LDM. They can probe an unexplored dark sector that is interacting with the standard model (SM) through one or more portals. At the lowest-beam-energy end, the DarkMESA experiment will run behind the dump of the 150-MeV electron beam of the MESA accelerator, currently under construction at the Institute for Nuclear Physics in Mainz. The concept for detecting direct scattering reactions of LDM comprises an electromagnetic calorimeter surrounded by an active veto system for rejecting backgrounds from SM particles. Suitable shielding will be located between the downstream detectors and the dump. A low-pressure, negative-ion, time-projection chamber could supplement these searches. At much higher beam energies, the Beam Dump eXperiment (BDX) is proposed to run parasitically behind the Jefferson Lab Hall-A beam dump making use of the up to 11-GeV electron beam. BDX employs the same detector concept. Direct LDM scattering reactions can be detected in an electromagnetic calorimeter operated inside hermetic layers of veto counters and a thick lead vault. Both experiments can explore uncovered regions of the parameter space of the LDM interaction strength versus mass, exceeding the discovery potential of existing experiments.