Yeah, so SCP/SFTP copy, not move (but you can delete the copy from the original computer to simulate that, which is your plan, I assume). Here's my solution:
I have a central SSH server where I store all my files. When I want a file on a given device, I pull it down from there. If I want to add a new file or update a file, I push a file up. Pretty simple. You do need at least one computer that stays in a constant location (typically at your house) to do this. From there, you portforward SSH to a high numbered port (it keeps every bot in the world from knocking, and some ISPs block 22) on your router. However, if you're like most of us, your ISP doesn't assign you a static IP. But, because your IP probably changes only rarely (probably only sometimes if your router goes down), you actually needn't use dynamic DNS. Furthermore, because you aren't actually publishing on that address, you can use any domain, regardless of ugliness, and there are plently of DNS services willing to offer you a subdomain for free. I reccomend freedns.afraid.org, myself.
It doesn't work for everyone, likely, but it works well enough for me.