When $B$ is $\acute{\rm e}$tale over $C$ and $A$ or $B$ is finite over $C$, 
then the result is known by Theorem (2.4) in my paper 
Descent for the $K$-theory of polynomial rings, [link text][1]


  [1]: http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002431580