  Consider a haploid population which has evolved through an exchangeable
reproduction dynamics, and in which all individuals alive at time $t$ have a
most recent common ancestor (MRCA) who lived at time $A_t$, say. As time goes
on, not only the population but also its genealogy evolves: some families will
get lost from the population and eventually a new MRCA will be established. For
a time-stationary situation and in the limit of infinite population size $N$
with time measured in $N$ generations, i.e. in the scaling of population
genetics which leads to Fisher-Wright diffusions and Kingman's coalescent, we
study the process $\mathcal A = (A_t)$ whose jumps form the point process of
time pairs $(E,B)$ when new MRCAs are established and when they lived. By
representing these pairs as the entrance and exit time of particles whose
trajectories are embedded in the look-down graph of Donnelly and Kurtz (1999)
we can show by exchangeability arguments that the times $E$ as well as the
times $B$ from a Poisson process. Furthermore, the particle representation
helps to compute various features of the MRCA process, such as the distribution
of the coalescent at the instant when a new MRCA is established, and the
distribution of the number of MRCAs to come that live in today's past.
