  Understanding the ultimate rate at which information propagates is a pivotal
issue in nonequilibrium physics. Nevertheless, the task of elucidating the
propagation speed inherent in quantum bosonic systems presents challenges due
to the unbounded nature of their interactions. In this Letter, we tackle the
problem of particle transport in long-range bosonic systems through the lens of
both quantum speed limits and the Lieb-Robinson bound. Employing a unified
approach based on optimal transport theory, we rigorously prove that the
minimum time required for macroscopic particle transport is always bounded by
the distance between the source and target regions, while retaining its
significance even in the thermodynamic limit. Furthermore, we derive an upper
bound for the probability of observing a specific number of bosons inside the
target region, thereby providing additional insights into the dynamics of
particle transport. Our results hold true for arbitrary initial states under
both long-range hopping and long-range interactions, thus resolving an open
problem of particle transport in generic bosonic systems.
