When $B$ is étale 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](http://www.digizeitschriften.de/id/266833020_0191|log39?tify={%22pages%22:[412,413],%22view%22:%22%22}&origin=/search?q%3Ddescent%2520for%2520the%2520k-theory).