Beyond Standard Model Physics at the LHeC and the FCC-eh

Thanks to the low background, negligible pileup and generally clean environement at 4? colliders, a rich program of physics beyond the Standard Model (BSM) can be explored with precision measurements, complementing searches at LHC and future ?? colliders. We review some recent studies aimed at evaluating the reach and sensitivity of the next generation 4? colliders, the LHeC and FCC-eh, in probing a few BSMmodels, in particular compressed scenarios of supersymmetry, long-lived particles and dark sector, charged Higgs and leptoquarks.