  We study piecewise-linear and birational lifts of Sch\"utzenberger promotion,
evacuation, and the RSK correspondence defined in terms of toggles. Using this
perspective, we prove that certain chain statistics in rectangles shift
predictably under the action of these maps. We then use this to construct
piecewise-linear and birational versions of Rubey's bijections between fillings
of equivalent moon polyominoes that preserve these chain statistics, and we
show that these maps form a commuting diagram. We also discuss how these
results imply Ehrhart equivalence and Ehrhart quasi-polynomial period collapse
of certain analogues of chain polytopes for moon polyominoes.
