While still dealing with this issue, I've stumbled upon this answer to a question asking about the conserved quantity corresponding to a scaling transformation. It mentions that in accordance with Noether's theorem, an improved energy-momentum tensor and Noether current can be found for a large class of (scale invariant) theories, such that the conserved charge can be calculated.
Unfortunately, apart from these short remarks, the OP of the answer I cite left no reference with further information about this issue. For example I'd like to know how such an improved energy-momentum tensor can be derived generally, what form it and the corresponding conserved charge would take for some example theories, how this conserved charge can be physically interpreted, etc.
Finally I'm interested in applying these ideas to fluid dynamics,I'd like to know how to construct the conserved quantity corresponding to the scale invariance of the Navier Stokes equations for example. But references wherein this concept is explained dealing with QFTs I'd appreciate too :-).