Hydrodynamicson embedded surfaces is useful for describing interesting physical systems suchas higher-dimensional black holes. When the fluid is in a stationary state onecan write an effective action for the fluid brane. Brane effective actions, ingeneral, have been useful in the description of the dynamics of strings andD-branes. Fine structure corrections to these actions have also been consideredin order to obtain improved effective actions for QCD or to probe the dynamicsof long strings. The dynamics of the probe branes arising from such effectivedescriptions is elastic in nature and its range of applicability is far widerthan string theory, describing the dynamics of fluid membranes studied intheoretical biology as well as the long-wavelength perturbations of blackbranes in generic gravity theories. In this talk we show how to iterativelycorrect brane effective actions for both extremal and non-extremal branes bytaking into account spin as well as curvature (elastic) corrections to thebrane dynamics. We then give examples of these effective theories in gravity bymeasuring the several response coefficients that characterize it, namely theYoung modulus and the piezoelectric moduli, from bent black branes via theblackfold approach. Finally, we also consider non-stationary fluidconfigurations and construct the theory of dissipative fluids living onembedded surfaces based on an analysis of the entropy current.