The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. Under a general outflow constraint, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. The main novelties of our approach are the construction of a flux carrier satisfying a uniform Leray-Hopf inequality in rectangular sections and the proof of some properties of weak solutions to the stationary Euler equations in bounded planar domains, such as the regularity of the extension to the whole plane, of the related Bernoulli pressure and of the stream function. This regularity is used to obtain local Morse-Sard-type information and to generate a solution through the invading domains procedure. For small data of the problem, we also prove unique solvability and attainability of Couette-Poiseuille flows at infinity. Applications of the results to the stability of suspension bridges are also given.