  We study various properties of composition operators acting between
generalized Fock spaces $\mathcal{F}_\varphi^p$ and $\mathcal{F}_\varphi^q$
with weight functions $\varphi$ grow faster than the classical Gaussian weight
function $\frac{1}{2}|z|^2$ and satisfy some mild smoothness conditions. We
have shown that if $p\neq q,$ then the composition operator $C_\psi:
\mathcal{F}_\varphi^p \to \mathcal{F}_\varphi^q $ is bounded if and only if it
is compact. This result shows a significance difference with the analogous
result for the case when $C_\psi$ acts between the classical Fock spaces or
generalized Fock spaces where the weight functions grow slower than the
Gaussian weight function. We further described the Schatten
$\mathcal{S}_p(\mathcal{F}_\varphi^2)$ class, normal, unitary, cyclic and
supercyclic composition operators. As an application, we characterized the
compact differences, the isolated and essentially isolated points, and
connected components of the space of the operators under the operator norm
topology.
