You can connect to the remote machine using ssh with X11 forwarding (-X) enabled. Then any graphical programs you run on the remote machine will have their output forwarded to the local display.
Code:
ssh -X -l raincityrunner othermachine.example.org
firefox &
That can be condensed further if you only want to run one program and then exit afterwards.
Code:
ssh -X -l raincityrunner othermachine.example.org firefox
You'll want a fast connection with low latency because it can be a little sluggish otherwise. Usually it is fine over the LAN.
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