= Solution
Use primes for conformal-time derivatives. The cold-dark-matter <Cosmological Euler equation in synchronous gauge> is $v_C'+\mathcal Hv_C=0$, so multiplying by $a$ gives $(av_C)'=0$. Therefore
$$
\boxed{v_C(\tau,x)=\frac{a(\tau_i)}{a(\tau)}v_C(\tau_i,x).}
$$
In an expanding universe $a$ increases, damping any initial <peculiar velocity> as $a^{-1}$. During <radiation domination> $a\propto\tau$, so the <velocity> scales as $\tau^{-1}$; during <matter domination> $a\propto\tau^2$, it scales as $\tau^{-2}$. This is the <nonrelativistic limit> of the physical-momentum <redshift> derived in Q1. It concerns the bulk <velocity> in the synchronous coordinate frame, not a growing <peculiar velocity> sourced in another gauge. A cold-matter rest-frame choice also removes the residual synchronous time displacement relevant to that component. The decay statement assumes expanding evolution and the pressureless, linear approximation used by the equation.
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