For a fixed positive integer $n\geq2$, basic properties of product $C_{\phi_n}\cdots C_{\phi_1}$ in $L^2(\mu)$ are presented, including the boundedness, adjoint and polar decomposition. Under the assistance of these properties, normality and quasinormality of bounded $C_{\phi_n}\cdots C_{\phi_1}$ in $L^2(\mu)$ are characterized respectively, where $C_{\phi_1}, C_{\phi_2},\cdots, C_{\phi_n}$ are all densely defined. Moreover, the compact intertwining relations of integral-type operators and composition operators between the Bloch-type spaces are presented.