  We consider a continuous-time random walk model with finite-mean
waiting-times and we study the mean first-passage time (MFPT) as estimated by
an observer in a reference frame $\mathcal{S}$, that is co-moving with a
target, and by an observer in a reference frame $\mathcal{S}'$, that is in
uniform motion with respect to the target through a Galilean transformation. We
found that the simple picture emerging in $\mathcal{S}$, where the mean
first-passage time depends on the whole jump distribution but only on the mean
value of the waiting-times, does indeed not hold in $\mathcal{S}'$ where the
estimation depends on the whole jump distribution and also on the whole
distribution of the waiting-times. We derive the class of jump-size
distributions such that the dependence of the MFPT on the mean waiting-time
only is conserved also in $\mathcal{S}'$. However, if the MFPT is finite, the
dependence on the specific waiting-time distribution disappears in
$\mathcal{S}'$ when the initial position is sufficiently far-away from the
target. While the MFPT emerges to be Galilean invariant with both two-sided and
one-sided jump distributions with finite moments, the MFPT is not a Galilean
invariant for one-sided jump distribution with power-law tails (one-sided
L\'evy distributions).
