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I was trying to go the other way as I downloaded calibre to be the repository for the books I've got from Amazon, so I've got a copy somewhere other than in the cloud. But haven't found a way of automating the emailing a copy to yourself as you have to do each book individually, I'm upto 130 odd and counting and it's a bit of a ball ache doing them one by one.
edit: googling seems to say now I can download direct from my kindle or through the kindle mac app (which if reviews are to be taken at face value is buggy as all hell).
Also when bringing directly into calibre it strips out the DRM and you can choose what format you want it to be in..
Will investigate further..
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On Windows I plug my Kindle Paperwhite into the PC running Calibre and import the books from there. I use this plug-in to strip the DRM https://github.com/apprenticeharper/DeDRM_tools/releases and it works automatically when you import the title.
If all of your books aren't on your Kindle you can just select them all and deliver to the device on the Amazon Website https://www.amazon.co.uk/hz/mycd/myx#/home/content/booksAll/dateDsc/
Yes, Kindles are surprisingly open. So long as it's in the right format you can email any book to your Kindle and it will store it in their cloud to download and sync across all devices.
No reason to buy all your books from the Amazon store, there are other stores that are compatible or things such as project Gutenberg with out of copyright books.