I have to agree with MxxCon, I suspect that your router or internet provider is causing your site to site transfers to be redirected through your local internet and then back out.
When transferring with SFTP the file is first downloaded to your local machine and then uploaded to the destination server.
Only FTP allows direct site to site transfers and only if none of the routers/firewalls in-between interfere with it.
If you have SSH access on the servers you can create a tar.gz archive of all the files on the original and wget it on the new server.
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