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Susan,
Many thanks for the very useful advice. I was not intending to clean the boxes out, but just give them a thorough drying out in a warm dry room/space (I am thinking of the airing cupboard). Do you think it might be necessary to clean the boxes? I should note that the boxes were not badly contaminated - just two to five specimens in each box were in need of a clean up.

Following your advice, I think I will try and get as many specimens as I can into wooden store boxes and I will definitely be relocating all boxes to a ground floor/first floor, well ventilated room.

Chris and Susan - Thanks again for the very helpful advice.:)