I have just moved from Australia to New Zealand. My collection of about 2000 CDs and 1000 DVDs were packed by the removal company in the usual shipping boxes along with the rest of our stuff, which came over in a shipping container. Most of the CDs were in sleeves in two large storage cases (as I had got rid of the "jewel" cases long ago).
Everything made it here safely in this way. I don't think you would need to find a specialist company.
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