Composition operators: ${\displaystyle N_{\alpha }}$ to the Bloch space to ${\displaystyle Q_{\beta }}$

Let ${\displaystyle N_{\alpha }}$,B and Q_β be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and ${\displaystyle Q_{\beta }}$ are Möbius invariant, but ${\displaystyle N_{\alpha }}$ is not. We characterize, in function-theoretic terms, when the composition operator ${\displaystyle C_{\varphi }f=f◦\varphi }$ induced by an analytic self-map ϕ of the unit disk defines an operator ${\displaystyle C_{\varphi }:N_{\alpha }\to B}$, ${\displaystyle B\to Q_{\beta }}$, ${\displaystyle N_{\alpha }\to Q_{\beta }}$ which is bounded resp. compact.