  In this paper we are concerned with Triebel-Lizorkin-Morrey spaces
$\mathcal{E}^{s}_{u,p,q}(\Omega)$ of positive smoothness $s$ defined on
(special or bounded) Lipschitz domains $\Omega\subset\mathbb{R}^d$ as well as
on $\mathbb{R}^d$. For those spaces we prove new equivalent characterizations
in terms of local oscillations which hold as long as some standard conditions
on the parameters are fulfilled. As a byproduct, we also obtain novel
characterizations of $\mathcal{E}^{s}_{u,p,q}(\Omega)$ using differences of
higher order. Special cases include standard Triebel-Lizorkin spaces $F^s_{p,q}
(\Omega)$ and hence classical $L_p$-Sobolev spaces $H^s_p(\Omega)$.
  Key words: Triebel-Lizorkin-Morrey space, Morrey space, Lipschitz domain,
oscillations, higher order differences
