  We investigate the ability of basis function expansions to reproduce the
evolution of a Milky Way-like dark matter halo, extracted from a cosmological
zoom-in simulation. For each snapshot, the density of the halo is reduced to a
basis function expansion, with interpolation used to recreate the evolution
between snapshots. The angular variation of the halo density is described by
spherical harmonics, and the radial variation either by biorthonormal basis
functions adapted to handle truncated haloes or by splines. High fidelity orbit
reconstructions are attainable using either method with similar computational
expense. We quantify how the error in the reconstructed orbits varies with
expansion order and snapshot spacing. Despite the many possible biorthonormal
expansions, it is hard to beat a conventional Hernquist-Ostriker expansion with
a moderate number of terms ($\gtrsim15$ radial and $\gtrsim6$ angular). As two
applications of the developed machinery, we assess the impact of the
time-dependence of the potential on (i) the orbits of Milky Way satellites, and
(ii) planes of satellites as observed in the Milky Way and other nearby
galaxies. Time evolution over the last 5 Gyr introduces an uncertainty in the
Milky Way satellites' orbital parameters of $\sim 15$ per cent, comparable to
that induced by the observational errors or the uncertainty in the present-day
Milky Way potential. On average, planes of satellites grow at similar rates in
evolving and time-independent potentials. There can be more, or less, growth in
the plane's thickness, if the plane becomes less, or more, aligned with the
major or minor axis of the evolving halo.
