A saxophone's biggest enemy during a move is trapped moisture sitting in the body and neck, which speeds up pad wear and can start corrosion in the tone holes. Swabbing it dry and securing the neck before it goes in the case covers the real risk.
Pro tip: A small silica gel packet inside the case, away from direct contact with the pads, helps control residual humidity on longer moves without affecting the felt.
Watch out: Never leave a saxophone unswabbed and sealed in its case for a move; trapped moisture accelerates pad deterioration and can lead to mold growth on felt and leather pads.