Suppose I have a Cartesian square of morphisms of algebraic varieties over a field $K$ (apologies for the grotty diagram):
A & \to^\alpha & B \\
\downarrow^\beta & & \downarrow^\gamma \\
C & \to^\delta & D
so $A$ is the fibre product of $B$ and $C$ over $D$. Suppose also that all four of $A,B,C,D$ are proper curves, and all the morphisms are finite and flat.
Is it then true that the two maps $K(C)^\times \to K(B)^\times$ given by
$\alpha_* \beta^*$ and
$\gamma^* \delta_*$ coincide?
(I am sure this must be standard, but I'm not a geometer and I don't really know where to look in the literature for this sort of thing.)