What is the effect(s) of the kinetic decoupling of WIMPs after it has frozen out? After freeze-out, the WIMPs departed from chemical equilibrium and till today, it maintains a constant number density. That appears enough to account for the observed relic density. But we also assume that it has kinetically decoupled at a later time i.e., elastic scatterings between observed matter and WIMPs stopped. 
Question Does kinetic decoupling have left any observable signature (like freeze-out has left the observed relic abundance)?
 A: Good question!
The main effect of the dark matter staying in kinetic equilibrium (after chemical decoupling), is that DM maintains a high temperature (defining the DM temperature by its kinetic energy): 
$$T_\chi \equiv \frac{2}{3} \left\langle \frac{p_\chi^2}{2m_\chi} \right\rangle = T_\gamma,$$
where $T_\chi$ is the DM temperature and $T_\gamma$ is the temperature of the heat bath (typically the standard model). 
The temperature of the heat bath typically goes like $T_\gamma \sim 1/a$, while the temperature of DM after kinetic decoupling goes like $T_\chi \sim 1/a^2$ (this is because in cosmology $p \sim 1/a$). 
So depending on the temperature of kinetic decoupling for DM, the temperature of the DM will be different, e.g. later kinetic decoupling will lead to a higher temperature. 
The effect of this high temperature will be that DM-overdensities on small scales will not grow (or start growing later) due to the high temperature of the DM. So only modes that enter the horizon well after kinetic decoupling of DM will remain unsuppressed(see figure).

 Plot of comoving scale versus scale factor. The thick blue line corresponds to the comoving horizon, while the red area represents the region where structure formation is suppressed by late kinetic decoupling. 
The observable effect of this is an exponential suppression in the DM power spectrum at a length scale corresponding to DM halo masses of roughly: 
$$M_\text{cut} \sim 5 \times 10^{10} M_\odot \left( \frac{T_\text{kd}}{100 \text{ eV}} \right)^{-3},$$
where $T_\text{kd}$ is the temperature at which DM undergoes kinetic decoupling.
For most typical WIMP-models the kinetic decoupling temperature is of order $T_\text{kd} \gtrsim \text{MeV}$ leading to completely unobservable cutoff mass. 
If, however, kinetic decoupling happens at temperatures around a keV (referred to as late kinetic decoupling), the resulting cutoff will be on dwarf galaxy scales and can possibly help address small-scale problems like the missing satellites problem. For a nice review of such a possibility, of which I'm an author :), see: arxiv.org/abs/1603.04884
