  The Volterra-type integral operator plays an essential role in modern complex
analysis and operator theory. Recently, Chalmoukis \cite{Cn} introduced a
generalized integral operator, say $I_{g,a}$, defined by
$$I_{g,a}f=I^n(a_0f^{(n-1)}g'+a_1f^{(n-2)}g''+\cdots+a_{n-1}fg^{(n)}),$$ where
$g\in H(\mathbb{D})$ and $a=(a_0,a_1,\cdots,a_{n-1})\in \mathbb{C}^n$. $I^n$ is
the $n$th iteration of the integral operator $I$. In this paper, we introduce a
more generalized integral operators $I_{\mathbf{g}}^{(n)}$ that cover $I_{g,a}$
on the Bloch-type space $\mathcal{B}^{\alpha}$, defined by
$$I_{\mathbf{g}}^{(n)}f=I^n(fg_0+\cdots+f^{(n-1)}g_{n-1}).$$ We show the
rigidity of the operator $I_{\mathbf{g}}^{(n)}$ and further the sum
$\sum_{i=1}^nI_{g_i}^{N_i,k_i}$, where
$I_{g_i}^{N_i,k_i}f=I^{N_i}(f^{(k_i)}g_i)$. Specifically, the boundedness and
compactness of $\sum_{i=1}^nI_{g_i}^{N_i,k_i}$ are equal to those of each
$I_{g_i}^{N_i,k_i}$. Moreover, the boundedness and compactness of
$I^n((fg')^{(n-1)})$ are independent of $n$ when $\alpha>1$.
