Tensor product over another tensor product

Question

Let $R$ be a ring and let $A$ and $B$ be $R$-algebras. Construct $A\otimes_R B$, the $R$-algebra given by tensor product. Let then $C$ and $D$ be two $A\otimes_RB$-algebras. Clearly $C$ and $D$ are also $R$-algebras, via restriction of scalars. Is it true, at least under some flatness condition, that $C\otimes_R D\cong C\otimes_{A\otimes_R B}D$, canonically, as $R$-algebras?