Let $A$, $B$, and $C$ be commutative rings such that $A\otimes_C B$ makes sense. If $W_n(A\otimes_C B), W_n(A), W_n(C),$ and $W_n(B)$ are the length $n$ Witt vectors of the rings $A,B,C,$ and $A\otimes_C B$. Is it true that

$$ W_n(A\otimes_C B)\cong W_n(A)\otimes_{W_n(C)}W_n(B)? $$

It seems as though this is a sensible property for Witt vectors to have. The case I am particularly interested in is the case when $C$ is a field of characteristic $p$ (not necessarily perfect) and $A$ and $B$ are $k$-algebras, but any suggestions for the general case would be helpful as well.