We consider formulations of the functional renormalization-group (fRG) flow for correlated electronic systems with the dynamical mean-field theory as a starting point. We classify the corresponding renormalization-group schemes into those neglecting one-particle irreducible six-point vertices (with respect to the local Green''s functions) and neglecting one-particle reducible six-point vertices. The former class is represented by the recently introduced DMF JRG approach [31], but also by the scale-dependent generalization of the one-particle irreducible representation (with respect to local Green''s functions, 1PI-LGF) of the generating functional [20]. The second class is represented by the fRG flow within the dual fermion approach [16,32]. We compare formulations of the fRG approach in each of these cases and suggest their further application to study 2D systems within the Hubbard model.