"I wish I had started this process in a screen session" problem
Why is everyone here assuming all programs can be run in a screen session? I run emacs over X11 through a tunnel. This is far superior to emacs in a screen session.
I've solved this by buying FastX from StarNet. I run a server on Linux and display my X programs on my desktop at work. When I get home, where I run the client on my home desktop, I can connect to the same server and disconnect my work clients and reconnect them to my home machine. It's the perfect solution for me.
Because that's what mosh is designed for. Graphical programs are off-topic.
By the way, we have used xpra ("screen for X") with success in the past: https://www.xpra.org/