  The (left-)curtain coupling, introduced by Beiglb\"ock and the author is an
extreme element of the set of "martingale" couplings between two real
probability measures in convex order. It enjoys remarkable properties with
respect to order relations and a minimisation problem inspired by the theory of
optimal transport. An explicit representation and a number of further
noteworthy attributes have recently been established by Henry-Labord\`ere and
Touzi. In the present paper we prove that the curtain coupling depends
continuously on the prescribed marginals and quantify this with Lipschitz
estimates. Moreover, we investigate the Markov composition of curtain couplings
as a way of associating Markovian martingales with peacocks.
