  We predict that one- and two-dimensional self-bound quantum droplets, forming
in Bose-Einstein condensates in the presence of Lee-Huang-Yang (LHY) quantum
corrections to the mean-field energy, may demonstrate exceptional mobility in
periodic optical lattices and that they may exhibit considerable displacements
across the lattice, remaining dynamically stable, even under weak initial phase
kicks imparted to them. Mobility properties of quantum droplets are determined
by their internal structure and strongly depend on the number of particles in
them. We find that due to the peculiar effect of the LHY quantum corrections,
odd (i.e., on-site centered) and even (i.e., intersite-centered)
one-dimensional quantum droplets feature alternating mobility and immobility
bands closely corresponding to the regions, where translational perturbation
mode is unstable and stable, respectively. This picture becomes even richer in
two-dimensional case, where odd-odd, even-odd or even-even quantum droplets
also feature alternating mobility and immobility domains, and where,
surprisingly, the droplet may be mobile in one direction, but immobile in the
orthogonal direction. We link changes in mobility properties with multiple
intersections of energy $E(\mu)$ and norm $N(\mu)$ dependencies for droplets
with different internal structure.
