  Motivated by the research on upper bounds on the rate of quantum transport
for one-dimensional operators, particularly, the recent works of
Jitomirskaya--Liu and Jitomirskaya--Powell and the earlier ones of
Damanik--Tcheremchantsev, we propose a method to prove similar bounds in
arbitrary dimension. The method applies both to Schroedinger and to long-range
operators.
  In the case of ergodic operators, one can use large deviation estimates for
the Green function in finite volumes to verify the assumptions of our general
theorem. Such estimates have been proved for numerous classes of quasiperiodic
operators in one and higher dimension, starting from the works of Bourgain,
Goldstein, and Schlag.
  One of the applications is a power-logarithmic bound on the quantum transport
defined by a multidimensional discrete Schr\"odinger (or even long-range)
operator associated with an irrational shift, valid for all Diophantine
frequencies and uniformly for all phases. To the best of our knowledge, these
are the first results on the quantum dynamics for quasiperiodic operators in
dimension greater than one that do not require exclusion of a positive measure
of phases. Moreover, and in contrast to localisation, the estimates are uniform
in the phase.
  The arguments are also applicable to ergodic operators corresponding to other
kinds of base dynamics, such as the skew-shift.
