We report bio-inspired (from a hummingbird's tongue) liquid transport via elastocapillary interaction of a thin membrane with a liquid meniscus. A soft wedge-thin rectangular membrane forming a wedge with a rigid substrate and a flat thin rectangular membrane undergo large deformation while interacting with liquid menisci. The membrane deformation leads to the formation of confinement which in turn results in elastocapillary flow along the membrane length. A simple theoretical model based on the Euler Bernoulli law is used to predict the membrane deformation profiles, which compare well with that obtained from experiments. In the wedge case, the membrane surface and liquid are selected such that the Concus-Finn criterion is not satisfied to contrast the present case of elastocapillary flow from the typical corner flow reported in the literature. The meniscus location versus time studies indicated that the flow exhibits the typical Washburn regime with , except for a sudden increase in velocity at the end of the membrane length. The effects of membrane thickness and width, liquids and substrates were studied to determine the expression for the modified Washburn constant Wm in both the wedge and flat membranes. It was found that gravity plays a role for Bo > 0.94 and for Bo = 1.9, the effect of inclination angle on the flow was studied. The elastocapillary flow with thin membranes could open up an opportunity for a new area, namely "membrane microfluidics" or "lab on a membrane", for diagnostics and other applications. © 2017 The Royal Society of Chemistry.