  We investigate the effect of thermal fluctuations on the atomic and
electronic structure of a twisted MoSe$_{2}$/WSe$_{2}$ heterobilayer using a
combination of classical molecular dynamics and \textit{ab-initio} density
functional theory calculations. Our calculations reveal that thermally excited
phason modes give rise to an almost rigid motion of the moir\'e lattice.
Electrons and holes in low-energy states are localized in specific stacking
regions of the moir\'e unit cell and follow the thermal motion of these
regions. In other words, charge carriers surf phason waves that are excited at
finite temperatures. Small displacements at the atomic scale are amplified at
the moir\'e scale, which gives rise to significant surfing speeds. We also show
that such surfing survives in the presence of a substrate and disorder. This
effect has potential implications for the design of charge and exciton
transport devices based on moir\'e materials.
