Summary: The concept of local growth envelope (\(\mathcal E_{\text{LG}}A, u\)) of a quasi-normed function space \(A\) is applied to spaces of generalized smoothness \(B^{(s,\Psi)}_{pq} (\mathbb R^n)\) and \(F^{(s,\Psi)}_{pq} (\mathbb R^n)\) and it is shown that the influence of the function \(\Psi\), which is a fine tuning of the main smoothness parameter \(s\), is strong enough in order to show up in the corresponding growth envelopes.