There is such a diagram--- it is the flux of energy in the linearized gravitational field (if you use full GR, you get complications with defining the energy). Unlike the electromagnetic case, where the electric field carry the bulk of the energy and the momentum of the charge carriers is negligible, in the gravitational case, it is the opposite.

You can also imagine electromagnetic circuits in which you accelerate very massive spheres which are very lightly charged, and use these as current carriers, and in this case, the momentum of the current carriers will not be negligible.