Let $L/K$ be a finite Galois extension of nonarchimedean local fields, and let $A$ and $A^t$ be dual abelian varieties over $K$. Tate local duality tells us that $A^t(K)$ and $H^1(K, A)$ are Pontryagin dual locally compact Hausdorff abelian topological groups, and likewise over $L$. Under this duality, why does the dual of the restriction map $H^1(K, A) \rightarrow H^1(L, A)$ identify with the norm map $A^t(L) \rightarrow A^t(K)$? 

I am aware that this is "well-known" and that there are references that treat special cases, but I am interested in a proof or a reference in the stated generality.

Edit: Since $K$ is _any_ nonarchimedean local field, the Tate local duality pairing $A^t(K) \times H^1(K, A) \rightarrow H^2(K, \mathbb{G}_m)$ that I am using is defined using biextensions (as in III.7 of Milne's Arithmetic Duality Theorems).