Idk, when I was transferring Wolfgang and Tutu from New Town to Treehut, I had both animals in both towns in boxes, so Tutu was in boxes in NT and someone else was in boxes in Treehut. I've successfully transferred both to my town, even while the other animal was in boxes.
However there was one time a while ago, Hopper was in Moriyo and I wanted someone from New Town to move to Moriyo. That animal in New Town wasn't asking to move in Moriyo, so I went back then changed it to the next day, Hopper is gone now, went back to New Town and that animal moved in Moriyo.
So I think it actually depends on how many animals you have in town. If you have a full 10, then maybe you can't ask another animal from another town to move in, but if you have 8, then you can, even if they are in boxes. Not sure about 9.