  Muller's ratchet, in its prototype version, models a haploid, asexual
population whose size~$N$ is constant over the generations. Slightly
deleterious mutations are acquired along the lineages at a constant rate, and
individuals carrying less mutations have a selective advantage. The classical
variant considers {\it fitness proportional} selection, but other fitness
schemes are conceivable as well. Inspired by the work of Etheridge et al.
([EPW09]) we propose a parameter scaling which fits well to the
``near-critical'' regime that was in the focus of [EPW09] (and in which the
mutation-selection ratio diverges logarithmically as $N\to \infty$). Using a
Moran model, we investigate the``rule of thumb'' given in [EPW09] for the click
rate of the ``classical ratchet'' by putting it into the context of new results
on the long-time evolution of the size of the best class of the ratchet with
(binary) tournament selection, which (other than that of the classical ratchet)
follows an autonomous dynamics up to the time of its extinction.
  In [GSW23] it was discovered that the tournament ratchet has a hierarchy of
dual processes which can be constructed on top of an Ancestral Selection graph
with a Poisson decoration. For a regime in which the mutation/selection-ratio
remains bounded away from 1, this was used in [GSW23] to reveal the asymptotics
of the click rates as well as that of the type frequency profile between
clicks. We will describe how these ideas can be extended to the near-critical
regime in which the mutation-selection ratio of the tournament ratchet
converges to 1 as $N\to \infty$.
