Fiddle leaf figs are famous for sulking after any disruption, dropping leaves and looking dramatic for a couple of weeks. You cannot prevent all leaf drop, but staking the trunk and sleeving the leaves keeps physical damage to a minimum.
Pro tip: Rotate the plant to face the same compass direction it faced in your old home; fiddle leaf figs are sensitive to light-direction changes.
Watch out: Do not carry a fiddle leaf fig by the trunk or a leaf; both snap or tear more easily than they look like they should.