BSM physics at the LHeC and the FCC-he

Electron-proton ($e^-p$) colliders are an ideal laboratory to study common features of electron and quarks with production via electroweak bosons, leptoquarks, multi-jet final states and very forward physics, due to their impressive pseudorapidity coverage. In addition to these physics cases, there exist a broad Beyond the Standard Model (BSM) program aimed at exploring the capabilities of the LHeC [1] and FCC-he [2] for several New Physics scenarios. Although their centre-of-mass energy is down with respect to a $pp$ collider by a factor of $\sqrt{E_p/E_e} \sim 10~(30)$ for the LHeC (FCC-he), they can be an invaluable tool to characterize BSM physics hints at $ee$ and $pp$ machines. The aim of this talk is to provide, on behalf of the BSM $e^-p$ Working Group, an overview of the aforementioned BSM program, by briefly summarizing the existing studies and reporting on the most recent progress. We expect that the ample scope in terms of NP models to be tested would enhance the synergies between the BSM and $e^-p$ communities