Fur is a natural material that reacts badly to heat, dry air, and prolonged plastic wrapping, all of which are common conditions inside a moving truck or a summer car trunk. Cool, dry, and breathable is the standard to protect it during transport.
Pro tip: If your move spans a long distance or a hot season, ask a local furrier about short-term cold storage rather than risking the coat in transit conditions.
Watch out: Sealed plastic dry-cleaning bags trap heat and humidity against fur, which dries and damages the fibers and the leather backing underneath.