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When a transaction is initited (for example transfer of ethers across accounts), does it first happen on the local copy of the blockchain and then synced over the network to other nodes? Or does the transaction look for the destination account on the blockchain of some other node (randomly) on the network?

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The transaction is transmitted to connected peers. Eventually it ends up with mining peers. Miners keep it in their pool and choose to include the transaction in a new block.

When the transaction is in a block, the whole network accepts it.

You should review your understanding of blockchain because there is no "looking for the destination account".

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When you initiate the transaction it is broadcasted to the other nodes (nearest one first), when it is verified they relay it to the others.
it arrives to the miners who add it to their own pools, then you will wait your transaction get mined in order to add it to the blockchain. So when it is added to a block this one in turn will be brodcasted to the whole network to be added to the current blockchain.

So you don't add anything locally and you don't send your transaction directly to the receiver's blockchain copy, There is no blockchain selection.

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