Let $X$ be a normal $\mathbb Q$-factorial variety (irreducible) over an algebraically closed field $k$ of characteristic $0$. Let $D\subseteq X$ be an irreducible divisor (which must be $\mathbb Q$-Cartier), $Z\subseteq X$ a closed subvariety, and $C\subseteq Z\subseteq X$ an irreducible curve.
We have the divisor-curve pairing $D\cdot C$ and the "higher cycle pairing" $D\cdot Z$ in the Chow ring of $X$. My question is the following:
Do we have $D\cdot C=(D\cdot Z)\cdot C$?
Here, $D\cdot C\in\mathbb Q$ is computed on $X$, and $(D\cdot Z)\cdot C\in\mathbb Q$ is a divisor-curve pairing computed on $Z$. My understanding of intersection theory is very minimal, so I'm not totally sure that this even makes sense.
The motivation for this question is the following. For $X$ a toric variety, $D$ a torus invariant divisor, and $Z$ a torus invariant subvariety, Lemma 12.5.2 of the "Toric varieties" book by Cox, Little, and Schenck tells us that $D\cdot Z=mD'$ where $D'=D\cap Z$ and $m$ is some multiplicity that can be computed using the fan data. Then I want to say that $$D\cdot C = (D\cdot Z)\cdot C = mD'\cdot C$$ and I know what $D'\cdot C$ should be in my situation.