We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross-section available to the fluid motion during the dynamics, sometimes called collapsible tubes. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we derive the fully three dimensional equations of motion. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. Next, we derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions. Finally, we derive the variational discretization of the problem, in both spatial and temporal dimensions, which is achieved by а discretization of the elastic tube and back-to-labels map of the fluid motion. The results of the experiments are also presented and discussed.