  We present a theoretical analysis of the dynamics of a one-dimensional
spin-1/2 fermionic gas subject to weak two-body losses. Our approach highlights
the crucial role played by spin conservation in the determination of the full
time evolution. We focus in particular on the dynamics of a gas that is
initially prepared in a Dicke state with fully-symmetric spin wavefunction, in
a band insulator, or in a Mott insulator. In the latter case, we investigate
the emergence of a steady symmetry-resolved purification of the gas. Our
results could help the modelisation and understanding of recent experiments
with alkaline-earth(-like) gases like ytterbium or fermionic molecules.
