A prior Q&A by Jellybean and Lior is instructive on why moving qubits in the surface code works via the hole-based method described in Fowler et al. I'm hoping for more details in why not only the logical operations can transfer, but also that the logical states are preserved, even after the $M_X$ measurement.
Namely, suppose we have some double defect Z qubit in a quiescent state $| \psi \rangle$ with three stabilizers of interest: $Z_0$ is the top stabilizer, $Z_1$ is the middle stabilizer, and $Z_2$ is the new stabilizer to move the hole to. Call the state post $M_X$ and $Z_1$ measured stabilizer to be $| \psi' \rangle$.
To demonstrate that the movement worked, it seems we must show that $\langle \psi' | Z_2 | \psi' \rangle = \langle \psi | Z_1 | \psi \rangle$, i.e. measuring the qubit would still retain the logical information (with some +/- factors depending on our intermediate phases). However, this equality seems non-trivial to prove -- so, how do we demonstrate the logical information is not only preserved but also extractable via simple $Z_2$ stabilizer measurement?