In this talk, we focus on an infinite-dimensional model of repulsively interacting Brownian motions: Dyson Brownian motion (DBM) at soft-edge scaling. It is known that its stationary process is the Airy line ensemble, a collection of non-intersecting random curves linked to many models in the KPZ universality class. We show that its time-marginal law is characterised as a Wasserstein steepest gradient descent (EVI gradient flow) of the relative entropy in the space of probability measures over the configuration space, an infinite-dimensional analogue of Jordan-Kinderlehrer-Otto/Ambrosio-Gigli-Savaré theory. We will discuss several applications such as RCD property, Varadhan short-time asymptotic and dynamical number-rigidity.