I always tie a piece of string to cables that need pulling through frames like this .If the cable is routed through the frame you'll find it easier to do if you cut the old display wire off and gaffer tape the new display wire to it and carefully use it to pull the new cable through the frame.
I have to ask how you propose to solder factory made looms together with factory fitted plugs as the displays dont come with bare wiresIf you need to pull new wires through the frame it would be safest to solder the new and old wires together, put some insulating tape around the joint to stop it snagging, and pull out the old wires.