In (Gottesman and Chuang 1999), when discussing quantum gate teleportation, they mention how it can be used to implement nonlocal gates such as a CNOT, by only using (classically controlled) local operations. Their scheme is shown in Figure 2 in the paper, which I report below:
Now, specifics of this particular circuit aside, I'm trying to get a better hold at the mechanism allowing nonlocal operations to be implemented via local operations and classical communication.
In general terms, I understand gate teleportation as performing state teleportation via the entangled state $(I\otimes U)|\Psi\rangle$ with $|\Psi\rangle\equiv\sum_i|i,i\rangle$ "standard" (symmetric) maximally entangled state. This results in a state $|\psi\rangle$ evolving to $(UU_a^\dagger U^\dagger)U|\psi\rangle$, conditionally to the ancilla being projected onto $|\Psi_a\rangle\equiv(U_a\otimes I)|\Psi\rangle$.
In a setting like the one in the figure above, the initial resource state $|\Psi\rangle$ oughts to become a four-qubit maximally entangled state $|\chi\rangle$. Somehow, performing Bell measurements on both ancillae (separately), and performing local operations on the state conditioned to these, results in a nonlocal operation being implemented. Now, I could obviously just follow the steps in the circuit above and verify that it indeed results in the CNOT being implemented, but my question concerns whether there is a more general/abstract way to understand this procedure. In particular, how is it that performing only local operations on the ancillae results in a nonlocal one on the target qubits?