  We consider randomized dynamics over the $n$-simplex, where at each step a
random set, or block, of coordinates is evenly averaged. When all blocks have
size 2, this reduces to the repeated averages studied in [CDSZ22], a version of
the averaging process on a graph [AL12]. We study the convergence to
equilibrium of this process as a function of the distribution of the block
size, and provide sharp conditions for the emergence of the cutoff phenomenon.
Moreover, we characterize the size of the cutoff window and provide an explicit
Gaussian cutoff profile. To complete the analysis, we study in detail the
simplified case where the block size is not random. We show that the absence of
a cutoff is equivalent to having blocks of size $n^{\Omega(1)}$, in which case
we provide a convergence in distribution for the total variation distance at
any given time, showing that, on the proper time scale, it remains constantly 1
up to an exponentially distributed random time, after which it decays following
a Poissonian profile.
