  We study embeddings between generalised Triebel-Lizorkin-Morrey spaces
${\mathcal E}^{s}_{\varphi,p,q}({\mathbb R}^d)$ and within the scales of
further generalised Morrey smoothness spaces like ${\mathcal
N}^{s}_{\varphi,p,q}({\mathbb R}^d)$, ${B}_{p,q}^{s,\varphi}({\mathbb R}^d)$
and ${F}_{p,q}^{s,\varphi}({\mathbb R}^d)$. The latter have been investigated
in a recent paper by the first two authors (2023), while the embeddings of the
scale ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$ were mainly obtained in a
paper of the first and last two authors (2022). Now we concentrate on the
characterisation of the spaces ${\mathcal E}^{s}_{\varphi,p,q}({\mathbb R}^d)$.
  Our approach requires a wavelet characterisation of those spaces which we
establish for the system of Daubechies' wavelets. Then we prove necessary and
sufficient conditions for the embedding ${\mathcal
E}^{s_1}_{\varphi_1,p_1,q_1}({\mathbb R}^d)\hookrightarrow {\mathcal
E}^{s_2}_{\varphi_2,p_2,q_2}({\mathbb R}^d)$. We can also provide some almost
final answer to the question when ${\mathcal E}^{s}_{\varphi,p,q}({\mathbb
R}^d)$ is embedded into $C({\mathbb R}^d)$, complementing our recent findings
in case of ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$.
