  We analyse the possibilities for the study of inclusive diffraction offered
by future electron--proton/nucleus colliders in the TeV regime, the Large
Hadron-electron Collider as an upgrade of the HL-LHC and the Future Circular
Collider in electron-hadron mode. Compared to $ep$ collisions at HERA, we find
an extension of the available kinematic range in $x$ by a factor of order $20$
and of the maximum $Q^2$ by a factor of order $100$ for LHeC, while the FCC
version would extend the coverage by a further order of magnitude both in $x$
and $Q^2$. This translates into a range of available momentum fraction of the
diffractive exchange with respect to the hadron ($\xi$), down to
$10^{-4}-10^{-5}$ for a wide range of the momentum fraction of the parton with
respect to the diffractive exchange ($\beta$). Using the same framework and
methodology employed in previous studies at HERA, considering only the
experimental uncertainties and not those stemming from the functional form of
the initial conditions or other ones of theoretical origin, and under very
conservative assumptions for the luminosities and systematic errors, we find an
improvement in the extraction of diffractive parton densities from fits to
reduced cross sections for inclusive coherent diffraction in $ep$ by about an
order of magnitude. For $eA$, we also perform the simulations for the Electron
Ion Collider. We find that an extraction of the currently unmeasured nuclear
diffractive parton densities is possible with similar accuracy to that in $ep$.
