The following thread also seems to be related to this issue.
As commented in the thread, August 4 update changed that N-Half Scape Dolls can now be stored in the Material Storage (https://ngs.pso2.com/news/updates/announcement62862).
N-Half Scape Dolls can now be stored in Material Storage.
*To use an N-Half Scape Doll, it must be in your Inventory.
Also fortunately, global site had the translated version of the announcement HarmlessSyan mentioned (https://ngs.pso2.com/news/announcements/accampaogn202106172). This announcement is talking about N-Scape Dolls, not N-Half Scape Dolls.
We have received reports that N-Scape Dolls are sometimes disappearing from players’ Inventories.
Upon further investigation, we could not confirm that N-Scape Dolls disappear.
However, players are able to store N-Scape Dolls in their Material Storage, so please check your Material Storage if your N-Scape Dolls have disappeared.
Interestingly, they are not saying this as a bug. They investigated to conclude that there's no bug that makes N-Scape Dolls disappear, but they said players might find their N-Scape Dolls which was lost from their inventory in their Material Storage.
So probably, N-Scape Dolls has been handled in the same way from the launch. Can be stored in the Material Storage, and moved to there by the batch move. Now N-Half Scape Dolls are handled in the same way after the last update.
I personally don't use batch move, but feel the current behavior is definitely faulty.