I’m no expert, but…
In situations like this I use the FlashFXP FTP client to transfer files directly from one server to another, without having to download them to my system first.
Many other FTP clients support the FXP protocol, so you shouldn’t have too many problems finding a suitable one. For more information on the FXP protocol, see the link below.
Edit: Another option is to log into your server using SSH and use the WGET utility to transfer files from a remote server. See the link below for more information.
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