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My wallet is not protected with a password, I have a few receiving addresses generated and a some funds in it, then I make a copy of the wallet and I set a password on this one.

Will the old unencrypted copy of the wallet still be able to spend current and future incoming funds without requiring a password ?

Or does something change in the private keys and addresses that gets updated with the network and old wallet becomes useless ?

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The old wallet is unaffected, assuming it uses separate data files (e.g. when new wallet is on a different computer or is a different software product).

The password is typically used to protect the program's critical data by encrypting it. It does not change the private-keys.

In the case of Bitcoin core, the data files would be wallet.dat (where private keys and other settings are stored) not the directories holding the blockchain data.

See https://en.bitcoin.it/wiki/Wallet_encryption

This page describes the algorithm used for encrypting the wallet.dat file used in the original Bitcoin client.

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Wallet encryption uses AES-256-CBC to encrypt only the private keys that are held in a wallet.

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At runtime, the client loads the wallet as it normally would, however the keystore stores the keys in encrypted form. When the passphrase is required (to top up keypool or send coins) it will either be queried by a GUI prompt, or must first be entered with the walletpassphrase RPC command. This will change the wallet to "unlocked" state ...

Obviously, if you had two different wallet programs using the exact same single wallet.dat file, encryption by the new software would affect use of the old software

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