Considering the generalized integration operator $$eft(C_{ǎrphi,g}^nf\right)(z)=ıt_{0}^{z}f^{(n)}(ǎrphi(\xi))g(\xi) d\xi,$$ between two Zygmund type spaces, the essential norm of the operator will be estimated. Here $\varphi$ is an analytic self-map on $\mathbb{D}$, $n\in\mathbb{N}$ and $g\in H(\mathbb{D})$. As a result, a criteria for the compactness of the above operator is given in the paper.