First, you are dealing with cable shielding and not equipment grounding. Think of it that way and things get clearer.
This is typically done with an EDAC panel that gets designed, financed, built, and never used after a couple of months. I figure the cost of relay boxes will be 2-3x the cost of a panel because the terminations are not as simple and brainless. You don't have to think much until it is time to block.
Shielding depends on what you have now. How are shields tied currently? How I wire, the shields would be tied at all consoles and lifted at the opposite ends at equipment.
The bestest gold star way would be to switch the shields as well, and load unconnected tape machine inputs with 600Ω. I'm dealing with older stuff, and some things do not like to sit with open inputs. Especially in NYC. This would 100% mimic an EDAC panel with termination plugs for unused machine inputs. Anything less would have to be tested in situ first before scaling up. The first compromise would be linking all the CH1 shields together, CH2, and so on in a single box. You want to have single shielded twisted pairs connecting the equipment as best as possible. When you start to link shields between different equipment the quiet lo-noise floor balance can be disrupted.
If you are thinking of taking the existing wiring, cutting it, and wiring it to a box filled with relays don't do it. Just don't do it. Any serious place, like commercial or rockstar home studios would never do this. They have EDAC panels. Dusty EDAC panels.
Mike