Will a space traveller slow down due to space expansion? Photons of relic radiation loose their energy as they propagate through space. Will a space traveler loose their peculiar velocity as he travels through vast distances? Will he stop somewhere or still he would be able to reach the stars at unlimited distance without burning the engines?
 A: If the initial velocity greatly  exceeds the escape velocity for the local galaxy cluster, then (barring the probability of traveler being diverted by some clump of matter)  we can assume that the traveler moves in the FLRW universe.
The conserved quantities of such motion would be covariant spatial components of the 4-speed $u_i$. Thus we have the equation:
$$
\frac{\gamma\, V}{\gamma_0\, V_0} = \frac{a_0}{a},
$$
where $V$ is the velocity measured in the comoving frame, $\gamma=(1+V^2)^{-\frac12}$ is the Lorentz factor, $a$ is the scale factor, while the index 0 is used to designate all of the quantities at the start of motion (we use units with $c=1$). 
So in expanding universe $a$ increases and thus the velocity $V$ would decrease. The timescales during which this decrease would occur are comparable with the inverse of Hubble constant. In the ultrarelativistic limit $\gamma \gg 1$, $V \to 1$ we can recover the equation for the redshift by substituting $\gamma \, V \to h \nu $.
What will be the end result of such a journey depends greatly on the exact law of the  scale factor growth $a(t)$. 
If the universe continues to expand with ever increasing acceleration (similar to de Sitter space, what the current observations seem to suggest for our universe), then there is a limit on what the traveler can reach: if an object has large enough (for a given velocity) redshift now, the traveler would never reach it. (Note, that because the universe is expanding, the (comoving) distance between the current position of the traveler and its starting point will exponentially increase with time). 
For the radiation dominated universe (scale factor behaves according to $a\sim t^{1/2}$) though the velocity approaches zero with time it does that more slowly and the total distance traveled is infinite, the traveler can reach any point, however distant with any initial velocity (though the time required could be exponentially large).
