  We define a tracial analogue of the sequentially split $*$-homomorphism
between $C^*$-algebras of Barlak and Szab\'{o} and show that several important
approximation properties related to the classification theory of $C^*$-algebras
pass from the target algebra to the domain algebra. Then we show that the
tracial Rokhlin property of the finite group $G$ action on a $C^*$-algebra $A$
gives rise to a tracial version of sequentially split $*$-homomorphism from
$A\rtimes_{\alpha}G$ to $M_{|G|}(A)$ and the tracial Rokhlin property of an
inclusion $C^*$-algebras $A\subset P$ with a conditional expectation $E:A \to
P$ of a finite Watatani index generates a tracial version of sequentially split
map. By doing so, we provide a unified approach to permanence properties
related to tracial Rokhlin property of operator algebras.
