For any two state vectors, |u> and |v>, unitary evolution preserves the amplitude <u|v>:
<u|U^\dag U|v> = <u|v>.
<u|U^\dag U|v> = <u|v>.
OTOH, I can understand the argument that nothing ever happens. One reasonable requirement for something to happen is that it should not be able to "unhappen," which is why some are looking for the possibility of irreversible decoherence.
This is not unitary. Quantum theory has two parts: unitary evolution, and measurement processes. That's it.
Let u and v be qubits. Apply a Hadamard gate to u while leaving v unchanged. Then they have evolved independently (but unitarily): u according to H, and v according to I.