  Motivated by recent measurements of deposits of $^{60}$Fe on the ocean floor
and the lunar surface, we model the transport of dust grains containing
$^{60}$Fe from a near-Earth (i.e., within 100 pc) supernova (SN). We inject
dust grains into the environment of an SN remnant (SNR) and trace their
trajectories by applying a 1D hydrodynamic description assuming spherical
symmetry to describe the plasma dynamics, and include a rudimentary, 3D
magnetic field description to examine its influence on charged dust grains. We
assume the interstellar medium (ISM) magnetic fields are turbulent, and are
amplified by the SNR shock, while the SN wind and ejecta fields are negligible.
We examine the various influences on the dust grains within the SNR to
determine when/if the dust decouples from the plasma, how much it is sputtered,
and where within the SNR the dust grains are located. We find that
Rayleigh-Taylor instabilities are important for dust survival, as they
influence the location of the SN's reverse shock. We find that the presence of
a magnetic field within the shocked ISM material limits the passage of SN dust
grains, with the field either reflecting or trapping the grains within the
heart of the SNR. These results have important implications for {\it in situ}
$^{60}$Fe measurements, and for dust evolution in SNRs generally.
