FileZilla: Transfer file remote server to remote server from a local computer

Solution 1:

FileZilla does not allow any kind of transfer between two remote systems using any protocol.

SFTP protocol does not allow a direct transfer of files between two remote systems.

Possible workarounds are:

• If you have an SFTP access, you will typically (but not necessarily) have an SSH shell access too. So you can login to the A and initialize a direct transfer to B from shell.
• If you want to keep using GUI, you probably do not have any other option than a transfer via a local temporary copy. Some SFTP clients allow this seamlessly by drag&drop. For example my WinSCP. See Can I transfer files between two remote servers with WinSCP?