Stamps are paper, which means humidity and pressure are their two real enemies during a move. Keeping albums dry and unstacked under weight protects decades of collecting in about half an hour of care.
Pro tip: A few silica gel packets, the kind that come in shoe boxes and electronics packaging, are enough to protect a whole album box from humidity.
Watch out: Humidity trapped in a sealed box during a long move can cause foxing or mold spots on paper stamps and album pages.