We consider the system of two immiscible fluids on a thin elastic sheet, where the fluid interface intersects with the sheet at a contact line. We first study the static profiles of the interfaces by minimising the total energy of the system. Asymptotic solutions are obtained in the limits as the bending modulus of the sheet tends to infinity (stiff limit) and zero (soft limit), respectively. Then we consider the dynamical problem and derive a hydrodynamic model, particularly the boundary conditions, from generalised thermodynamics. Numerical solutions are presented for the relaxation of droplets on a membrane and transport of droplets by bendotaxis.