Can I just say that if your collection is the “real thing” & not costume jewellery that you take care with that lovely jewel case. IF you were burgled the perpetrator could simply lift the box and walk away.
If indeed your items are very precious, you might consider having a safe fitted. I have one & it does help peace of mind albeit I also have an intruder alarm.
Your local locksmith can probably help with a suitably sized safe & the fitting thereof. Make sure it’s big enough to take items within their boxes.
Meanwhile, if you choose to sell any items or your heirs likewise, original boxes (and certificates where appropriate) are invaluable to the provenance of an item.
Finally, if you can, may I suggest you take photos of your lovely items, in case of loss, to help insurance claims.
In your shoes, to answer your original query, I’d keep items in their boxes and alternate them into the jewel box whilst putting the other items away elsewhere out of sight & secured.
Hope this helps to clarify your thoughts.?