Afaik there is no way to sync just the newer blocks from one wallet to another.
It's calling "Syncing"... and it's what happens when your Bitcoin Core instance connects to peers 
I'm not sure what it is that the OP is actually trying to achieve... why is it necessary to copy blocks from Windows to Linux? Why not just hook up the Linux machine to the network and let it sync itself?
If this is supposed to be some sort of offline air-gapped Linux machine... you don't need the blocks on it anyway.
He didn't say this is an air-gap cold storage setup where space doesn't matter, he wants from what I can understand a regular full node in windows. The reason you would want to save the entire blockchain is that if you move the "blocks" folder (without the "chainstate" folder, otherwise it will become corrupted since the chainstate files need to be generated from scratch) you can save a lot of time by not downloading the blocks, you would only validate the existing blocks you kept. So always try to keep the entire blockchain somewhere in case you need to sync from scratch.
Ok, not the best description up there. Let's say to import the blocks from the own wallet on the windows machine.