  Using \Planck\ data combined with the Meta Catalogue of X-ray detected
Clusters of galaxies (MCXC), we address the study of peculiar motions by
searching for evidence of the kinetic Sunyaev-Zeldovich effect (kSZ). By
implementing various filters designed to extract the kSZ generated at the
positions of the clusters, we obtain consistent constraints on the radial
peculiar velocity average, root mean square (rms), and local bulk flow
amplitude at different depths. For the whole cluster sample of average redshift
0.18, the measured average radial peculiar velocity with respect to the cosmic
microwave background (CMB) radiation at that redshift, i.e., the kSZ monopole,
amounts to $72 \pm 60$ km s$^{-1}$. This constitutes less than 1% of the
relative Hubble velocity of the cluster sample with respect to our local CMB
frame. While the linear $\Lambda$CDM prediction for the typical cluster radial
velocity rms at $z=0.15$ is close to 230km s$^{-1}$, the upper limit imposed by
\Planck\ data on the cluster subsample corresponds to 800 km s$^{-1}$ at 95%
confidence level, i.e., about three times higher. \Planck\ data also set strong
constraints on the local bulk flow in volumes centred on the Local Group. There
is no detection of bulk flow as measured in any comoving sphere extending to
the maximum redshift covered by the cluster sample. A blind search for bulk
flows in this sample has an upper limit of 254 km s$^{-1}$ (95% confidence
level) dominated by CMB confusion and instrumental noise, indicating that the
Universe is largely homogeneous on Gpc scales. In this context, in conjunction
with supernova observations, \Planck\ is able to rule out a large class of
inhomogeneous void models as alternatives to dark energy or modified gravity.
The \Planck\ constraints on peculiar velocities and bulk flows are thus
consistent with the $\Lambda$CDM scenario.
