  Future Cosmic Microwave Background experiments together with upcoming galaxy
and 21-cm surveys will provide extremely accurate measurements of different
cosmological observables located at different epochs of the cosmic history. The
new data will be able to constrain the neutrino mass sum with the best
precision ever. In order to exploit the complementarity of the different
redshift probes, a deep understanding of the physical effects driving the
impact of massive neutrinos on CMB and large scale structures is required. The
goal of this work is to describe these effects, assuming a summed neutrino mass
close to its minimum allowed value. We find that parameter degeneracies can be
removed by appropriate combinations, leading to robust and model independent
constraints. A joint forecast of the sensitivity of Euclid and DESI surveys
together with a CORE-like CMB experiment leads to a $1\sigma$ uncertainty of
$14$~meV on the summed neutrino mass. However this particular combination gives
rise to a peculiar degeneracy between $M_\nu$ and the optical depth at
reionization. Independent constraints from 21-cm surveys can break this
degeneracy and decrease the $1\sigma$ uncertainty down to $12$~meV.
