Tried just once (the whole setup is rather new). I tried the first part of Hellblade running on my PC, which is in the same room, at 1440p & High settings. I played for about half an hour, with no noticeable glitches or lag. Both the PC and the Shield are connected using cables (each to a different switch, both switches go directly to the router).

I hope I have time to try it more before the weekend, but the results are promising.

Oh, I thought it was obvious since we're talking about a Shield Pro but now I think it's worth mentioning that my PC has a Nvidia card (an RTX2060), so I'm using nvidia's native streaming support. It's also worth noting that an OSS client exists (Moonlight), so the Shield could be replaced by, say, a Raspberry Pi 4