Once I started packing boxes, both of my cats got a little anxious - and because it was a fairly major move (1000 miles), I started packing about 2 months before the actual move. One kitty isn't usually easily stressed, but the other is super skittish. I just tried to give them lots of extra attention, and I left the room where they spent the most time as unchanged as possible for as long as possible. It was the last room I packed up. I made sure to shut them in one room while the others were being emptied and until the movers were gone. In that sort of stressful situation, cats might think first of fleeing. And doors are invariably going to be left open.
You could also try Feliway, or pheromone collars, or other things like essences to put in drinking water that are supposed to calm them.
I did the same when we arrived at the new place. Put them in one room with their litter boxes, food, and water, for a while. When the movers arrived, I put them in the bathroom.