Suppose $X$ is a scheme and $Y$ and $Z$ are closed subschemes such that $Y$ is a finite union of closed reduced irreducible subschemes $C$ that satisfy $$\mathscr{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_C,\mathcal{O}_Z)=0$$ for all $i>0$.
Does it necessarily follow that $$\mathscr{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_Y,\mathcal{O}_Z)=0$$
For all $i>0$?
We can assume that all sheaves above are coherent and that there are enough projectives so that we can define Tor.
If this isn't true, then could we say something if $Y$ is also reduced?
I'm not really sure exactly how to think about this, but I know that vanishing of Tor is a certain type of transversality, so it seems like there should be some sort of result of this flavor.