  We consider a Markovian evolution on point processes, the $\Psi$--process, on
the unit interval in which points are added according to a rule that depends
only on the spacings of the existing point configuration. Having chosen a
spacing, a new point is added uniformly within it. Building on previous work of
the authors and of Junge, we show that the empirical distribution of points in
such a process is always equidistributed under mild assumptions on the rule,
generalizing work of Junge.
  A major portion of this article is devoted to the study of a particular
growth--fragmentation process, or cell process, which is a type of
piecewise--deterministic Markov process (PDMP). This process represents a
linearized version of a size--biased sampling from the $\Psi$--process. We show
that this PDMP is ergodic and develop the semigroup theory of it, to show that
it describes a linearized version of the $\Psi$--process. This PDMP has
appeared in other contexts, and in some sense we develop its theory under
minimal assumptions.
