  We study composition-valued continuous-time Markov chains that appear
naturally in the framework of Chinese Restaurant Processes (CRPs). As time
evolves, new customers arrive (up-step) and existing customers leave
(down-step) at suitable rates derived from the ordered CRP of Pitman and Winkel
(2009). We relate such up-down CRPs to the splitting trees of Lambert (2010)
inducing spectrally positive L\'{e}vy processes. Conversely, we develop
theorems of Ray-Knight type to recover more general up-down CRPs from the
heights of L\'{e}vy processes with jumps marked by integer-valued paths. We
further establish limit theorems for the L\'{e}vy process and the
integer-valued paths to connect to work by Forman et al. (2018+) on interval
partition diffusions and hence to some long-standing conjectures.
