  Large low surface brightness galaxies have recently been found to be abundant
in nearby galaxy clusters. In this paper, we investigate these ultra-diffuse
galaxies (UDGs) in the six Hubble Frontier Fields galaxy clusters: Abell 2744,
MACSJ0416.1$-$2403, MACSJ0717.5$+$3745, MACSJ1149.5$+$2223, Abell S1063 and
Abell 370. These are the most massive ($1$-$3 \times 10^{15}~M_\odot$) and
distant ($0.308 < z < 0.545$) systems in which this class of galaxy has yet
been discovered. We estimate that the clusters host of the order of
${\sim}$200-1400 UDGs inside the virial radius ($R_{200}$), consistent with the
UDG abundance halo-mass relation found in the local universe, and suggests that
UDGs may be formed in clusters. Within each cluster, however, we find that UDGs
are not evenly distributed. Instead their projected spatial distributions are
lopsided, and they are deficient in the regions of highest mass density as
traced by gravitational lensing. While the deficiency of UDGs in central
regions is not surprising, the lopsidedness is puzzling. The UDGs, and their
lopsided spatial distributions, may be associated with known substructures late
in their infall into the clusters, meaning that we find evidence both for
formation of UDGs in clusters and for UDGs falling into clusters. We also
investigate the ultra-compact dwarfs (UCDs) residing in the clusters, and find
the spatial distributions of UDGs and UCDs appear anti-correlated. Around 15%
of UDGs exhibit either compact nuclei or nearby point sources. Taken together,
these observations provide additional evidence for a picture in which at least
some UDGs are destroyed in dense cluster environments and leave behind a
residue of UCDs.
