  We investigate the transport of inertial particles by cellular flows when
advection dominates over inertia and diffusion, that is, for Stokes and
P\'eclet numbers satisfying $\mathrm{St} \ll 1$ and $\mathrm{Pe} \gg 1$.
Starting from the Maxey--Riley model, we consider the distinguished scaling
$\mathrm{St} \, \mathrm{Pe} = O(1)$ and derive an effective Brownian dynamics
approximating the full Langevin dynamics. We then apply homogenisation and
matched-asymptotics techniques to obtain an explicit expression for the
effective diffusivity $\overline{D}$ characterising long-time dispersion. This
expression quantifies how $\overline{D}$, proportional to $\mathrm{Pe}^{-1/2}$
when inertia is neglected, increases for particles heavier than the fluid and
decreases for lighter particles. In particular, when $\mathrm{St} \gg
\mathrm{Pe}^{-1}$, we find that $\overline{D}$ is proportional to
$\mathrm{St}^{1/2}/(\log ( \mathrm{St} \, \mathrm{Pe}))^{1/2}$ for heavy
particles and exponentially small in $\mathrm{St} \, \mathrm{Pe}$ for light
particles. We verify our asymptotic predictions against numerical simulations
of the particle dynamics.
