  We define and examine sequentially split $*$-homomorphisms between
$\mathrm{C}^*$-algebras and $\mathrm{C}^*$-dynamical systems. For a
$*$-homomorphism, the property of being sequentially split can be regarded as
an approximate weakening of being a split-injective inclusion of
$\mathrm{C}^*$-algebras. We show for a sequentially split $*$-homomorphism that
a multitude of $\mathrm{C}^*$-algebraic approximation properties pass from the
target algebra to the domain algebra, including virtually all important
approximation properties currently used in the classification theory of
$\mathrm{C}^*$-algebras. We also discuss various settings in which sequentially
split $*$-homomorphisms arise naturally from context. One particular class of
examples arises from compact group actions with the Rokhlin property. This
allows us to recover and extend the presently known permanence properties of
Rokhlin actions with a unified conceptual approach and a simple proof.
Moreover, this perspective allows us to obtain new results about such actions,
such as a generalization of Izumi's original $K$-theory formula for the fixed
point algebra, or duality between the Rokhlin property and approximate
representability.
