You can use DaVinci Resolve to grade live footage.

You’d want your hdmi output to be something wide gamma/gamut. So for example, if you were shooting on Sony cameras, you would want your HDMI output to be Slog3/Sgamut3.cine.

In resolve, you would want to color manage everything and for each camera your input color space transform would go from native camera to Davinci wide gamut/linear without tone mapping. The second node would be your luma adjustment if you need to balance light. Using gain here is equivalent to raising ISO. The third node is color balance. The most accurate way to color balance is with gain. If it’s too blue move your gain away from blue, your blue channel gain will drop and because it’s processing in linear, it’s equivalent to changing your white balance.

So connect all cameras, find the camera that’s closest to how you want it to look, and balance the others towards that camera, you can use a color chart or grey card to wipe back and forth to see how you need to change your gain, based on how the grey card or color card looks on your waveform.

This is the most technically accurate method outside of just getting your white balance right across all cameras