  We study the renormalization group flow in weak power counting (WPC)
renormalizable theories. The latter are theories which, after being formulated
in terms of certain variables, display only a finite number of independent
divergent amplitudes order by order in the loop expansion. Using as a toolbox
the well-known SU(2) non linear sigma model, we prove that for such theories a
renormalization group equation holds that does not violate the WPC condition:
that is, the sliding of the scale $\mu$ for physical amplitudes can be
reabsorbed by a suitable set of finite counterterms arising at the loop order
prescribed by the WPC itself. We explore in some detail the consequences of
this result; in particular, we prove that it holds in the framework of a
recently introduced beyond the Standard Model scenario in which one considers
non-linear St\"uckelberg-like symmetry breaking contributions to the fermion
and gauge boson mass generation mechanism.
