  Given two probability measures on sequential data, we investigate the
transport problem with time-inconsistent preferences under a discrete-time
setting. Motivating examples are nonlinear objectives, state-dependent costs,
and regularized optimal transport with general $f$-divergence. Under the
bicausal constraint, we introduce the concept of equilibrium transport. The
existence is proved in the semi-discrete Markovian case and the continuous
non-Markovian case with strict quasiconvexity, while the uniqueness also holds
in the second case. We apply our framework to study inertia of two job markets,
top-ranking executives and academia. The empirical analysis shows that a job
market with stronger inertia is less efficient. The University of California
(UC) postdoc job market has the strongest inertia even than that of executives,
while there is no evidence of inertia in the UC faculty job market.
