There is a PiKVM project that does something like this for the raspberry pi, forwarding the KVM over IP. You could probably attach similar hardware to the framework 16, and hook it up to a framework 13 board over usb. The software that is used is open source and should run fine on the framework 16 on one of the supported Linux distro’s.
See here:
The hardware involved is quite substantial however, I don’t see a way to make this work in a expansion card or cable, but it would probably fit fine in a small external case.