Let $K/L/M$ be finite Galois function field extensions and let $P|Q|S$ be places of the corresponding function fields. Write $G_i(\cdot|\cdot)$ for the $i$-th ramification groups.
One can easily show that the canonical projection $Gal(M/K) \rightarrow Gal(L/K)$ projects $G_i(S|P)$ into $G_i(Q|P)$ surjectively for $i=-1,0$.
What about the higher $i$'s?
In general, this projection induces an embedding $G_i(S|P)/G_i(S|Q) \rightarrow G_i(Q|P)$ and this has to be an isomorphism for $i=-1,0$ because the cardinalities of the left and right side are the same. But this holds only since the ramification groups are the decompositon and inertia groups and their cardinalities are $ef(\cdot|\cdot)$ and $e(\cdot|\cdot)$, which again satisfy the multiplicative transitivity rule.
Thank you.