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We study unboundedness properties of functions belonging to generalisedMorrey spaces ${\mathcal M}_{\varphi,p}({\mathbb R}^d)$ and generalisedBesov-Morrey spaces ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$ by means ofgrowth envelopes. For the generalised Morrey spaces we arrive at the same threepossible cases as for classical Morrey spaces $\mathcal{M}_{u,p}({\mathbbR}^d)$, i.e., boundedness, the $L_p$-behaviour or the proper Morrey behaviourfor $p<u$, but now those cases are characterised in terms of the limit of$\varphi(t)$ and $t^{-d/p} \varphi(t)$ as $t \to 0^+$ and $t\to\infty$,respectively. For the generalised Besov-Morrey spaces the limit of $t^{-d/p}\varphi(t)$ as $t \to 0^+$ also plays a r\^ole and, once more, we are able toextend to this generalised spaces the known results for classical Besov-Morreyspaces, although some cases are not completely solved. In this context we cancompletely characterise the situation when ${\mathcalN}^{s}_{\varphi,p,q}({\mathbb R}^d)$ consists of essentially bounded functionsonly, and when it contains regular distributions only.

On a bridge connecting Lebesgue and Morrey spaces in view of their growth properties