this is just one thing that come on the top of my mind if you want to do the transfer via external hard drive or flash disk

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temporarily disconnect the source computer from the network, temporarily disable the antivirus, and start copying the files. disabling the antivirus should help the system copy files much faster. but of course you lose the protection of scanning the files as they are transferred to the external drive.

or you may want to boot into safe mode instead and copy the files to the external drive while in safe mode.

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try to temporarily disable the Windows 7 firewall OR set your LAN connection settings to the "Public" location (and not Home). Turning off the firewall should lessen the networking "restrictions" of Windows 7.

but try first with just setting the LAN connection to the Public location instead of the Home location.

goodluck