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In proper-time gauge the particle Lagrangian is $L=m(-{\dot{t}}^2+a^2{\dot{\mathbf{x}}}^2-1)/2$, now with dots referring to proper time $s$. The canonical spatial momentum isThe FRW metric is independent of all spatial coordinates. Hence $\partial L/\partial x^i=0$ and the Euler-Lagrange equation gives $d{p}_i^C/ds=0$. This conserved quantity is the comoving momentum.

The mass-shell mass shell constraint gives $(dt/ds{)}^2-a^2|d\mathbf{x}/ds{|}^2=1$. For future-directed motion and proper peculiar velocity $\mathbf{v}=ad\mathbf{x}/dt$, it follows that $dt/ds=(1-v^2{)}^{-1/2}=\gamma$. ThereforeThe kinetic momentum here is the physical momentum in an FRW universe, measured in the orthonormal frame of a comoving observer. It redshifts as $a^{-1}$ irrespective of whether the particle is relativistic or nonrelativistic.