  [Abridged] We present K-band data for the brightest cluster galaxies (BCGs)
from the ESO Distant Cluster Survey. These data are combined with photometry
from Aragon-Salamanca et al. (1998) and a low-redshift comparison sample from
von der Linden et al. (2007). The K-band Hubble diagram for BCGs exhibits very
low scatter (~0.35mag) since z=1. The colour and $K$-band luminosity evolution
of the BCGs are in good agreement with passively-evolving stellar populations
formed at z>2. We do not detect any significant change in the stellar mass of
the BCG since z~1. These results do not seem to depend on the velocity
dispersion of the parent cluster. There is a correlation between the 1D
velocity dispersion of the clusters and the K-band luminosity of the BCGs
(after correcting for passive evolution). The clusters with large velocity
dispersions tend to have brighter BCGs, i.e., BCGs with larger stellar masses.
This dependency, although significant, is relatively weak: the stellar mass of
the BCGs changes only by ~70% over a two-order-of-magnitude range in cluster
mass. This dependency doesn't change significantly with redshift. The models of
De Lucia & Blaizot (2007) predict colours which are in reasonable agreement
with the observations because the growth in stellar mass is dominated by the
accretion of old stars. However, the stellar mass in the model BCGs grows by a
factor of 3-4 since z=1, a growth rate which seems to be ruled out by the
observations. The models predict a dependency between the BCG's stellar mass
and the velocity dispersion of the parent cluster in the same sense as the
data, but the dependency is significantly stronger than observed. However, one
major difficulty in this comparison is that we have measured fixed metric
aperture magnitudes while the models compute total luminosities.
