  In a random complete and separable metric space that we call the lookdown
space, we encode the genealogical distances between all individuals ever alive
in a lookdown model with simultaneous multiple reproduction events. We
construct families of probability measures on the lookdown space and on an
extension of it that allows to include the case with dust. From this
construction, we read off the tree-valued $\Xi$-Fleming-Viot processes and
deduce path properties. For instance, these processes usually have a.s.
c\`adl\`ag paths with jumps at the times of large reproduction events. In the
case of coming down from infinity, the construction on the lookdown space also
allows to read off a process with values in the space of measure-preserving
isometry classes of compact metric measure spaces, endowed with the
Gromov-Hausdorff-Prohorov metric. This process has a.s. c\`adl\`ag paths with
additional jumps at the extinction times of parts of the population.
