  Establishing cutoff -- an abrupt transition from "not mixed" to "well mixed"
-- is a classical topic in the theory of mixing times for Markov chains.
Interest has grown recently in determining not only the existence of cutoff and
the order of its mixing time and window, but the exact shape, or profile, of
the convergence inside the window. Classical techniques, such as coupling or
$\ell_2$-bounds, are typically too crude to establish this and there has been a
push to develop general techniques.
  We build upon this work, extending from conjugacy-invariant random walks on
groups to certain projections. We exemplify our method by analysing the
$k$-particle interchange process on the complete $n$-graph with $k \asymp n$.
This is a projection of the random-transpositions card shuffle, which
corresponds to $k = n$, analysed by Teyssier.
