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We investigate duality and existence of dual optimizers for several adaptedoptimal transport problems under minimal assumptions. This includes the causaland bicausal transport, the barycenter problem, and a general multimarginalproblem incorporating causality constraints. Moreover, we discuss applicationsof our results in robust finance. We consider a non-dominated model of severalfinancial markets where stocks are traded dynamically, but the joint stockdynamics are unknown. We show that a no-arbitrage assumption in a quasi-suresense naturally leads to sets of multicausal couplings. Consequently, computingthe robust superhedging price is equivalent to solving an adapted transportproblem, and finding a superhedging strategy means solving the correspondingdual.

General duality and dual attainment for adapted transport