A large sample of Abell clusters of galaxies, selected for the likely presence of a dominant galaxy, is used to study the dynamical properties of the brightest cluster members (BCMs). From visual inspection of Digitized Sky Survey images combined with redshift information we identify 1426 candidate BCMs located in 1221 different redshift components associated with 1169 different Abell clusters. This is the largest sample published so far of such galaxies. From our own morphological classification we find that ~92% of the BCMs in our sample are early-type galaxies and 48% are of cD type. We confirm what was previously observed based on much smaller samples, namely, that a large fraction of BCMs have significant peculiar velocities. From a subsample of 452 clusters having at least 10 measured radial velocities, we estimate a median BCM peculiar velocity of 32% of their host clusters' radial velocity dispersion. This suggests that most BCMs are not at rest in the potential well of their clusters. This phenomenon is common to galaxy clusters in our sample, and not a special trait of clusters hosting cD galaxies. We show that the peculiar velocity of the BCM is independent of cluster richness and only slightly dependent on the Bautz-Morgan type. We also find a weak trend for the peculiar velocity to rise with the cluster velocity dispersion. The strongest dependence is with the morphological type of the BCM: cD galaxies tend to have lower relative peculiar velocities than elliptical galaxies. This result points to a connection between the formation of the BCMs and that of their clusters. Our data are qualitatively consistent with the merging-groups scenario, where BCMs in clusters formed first in smaller subsystems comparable to compact groups of galaxies. In this scenario, clusters would have formed recently from the mergers of many such groups and would still be in a dynamically unrelaxed state.