Where were the elements stored when you moved? In the household inventory or your Sim's personal inventory? If in the household inventory, did you drag them to the ground or a table, then in live mode try to drag from there to the display? Perhaps since you just moved, the game doesn't recognize them until put into your Sim's personal inventory first? Are you using the Sim that owns them to try to drag to the display rack? Are you clicked on a child or ghost Sim when trying to drag them?
I don't know if any of these things make a difference, or if you've already tried dragging them to the rack from different areas or with different Sims, but it's worth a try. I had the same problem with elements when my family moved. I don't recall exactly what I did to solve it. But I do remember the rack & elements were in the household inventory and I moved the elements to the new lot ground after placing the rack, then tried to drag the elements to the rack in live mode and they wouldn't go in it.
I think what I had to do was put them in my Sim's personal inventory and then drag them to the rack from there. If none of this works, you could try buying a new rack or maybe try to reset it? Just some ideas, I hope you get it figured out. :)
