  Continuous-time Mallows processes are processes of random permutations of the
set $\{1, \ldots, n\}$ whose marginal at time $t$ is the Mallows distribution
with parameter $t$. Recently Corsini showed that there exists a unique Markov
Mallows process whose left inversions are independent counting processes. We
prove that this process admits a global and a local limit as $n \to \infty$.
The global limit, obtained after suitably rescaling space and time, is an
explicit stochastic process on $[0,1]$ whose description is based on the
permuton limit of the Mallows distribution, analyzed by Starr. The local limit
is a process of permutations of $\mathbb{Z}$ which is closely related to the
construction of the Mallows distribution on permutations of $\mathbb{Z}$ due to
Gnedin and Olshanski. Our results demonstrate an analogy between the asymptotic
behavior of Mallows processes and the recently studied limiting properties of
random sorting networks.
