Updated macOS with OpenCore 0.6.8 on a number of systems, not just Big Sur. Three down 6 to go!
iMac1 - Big Sur 11.3
iMac1 - 10.15.7 Security Update 2021-002 Catalina
iMac6 - Big Sur 11.3
iMac6 - 10.15.7 Security Update 2021-002 Catalina
iMac6 - 10.14.6 Security Update 2021-003 Mojave
Mac-Server2 - 10.14.6 Security Update 2021-003 Mojave
Only one system so far has caused me any issues, my Mac-Server running Catalina, I tried updating the system remotely and it got stuck in an OC boot loop. I'll have to connect a screen and keyboard to see what is wrong with this one.
Edit:
Problem system wasn't caused by macOS update, but rather by a new CPU fan that wouldn't spin up to pass the HP Server Hardware Monitoring test. Swapped the old noisy HP fan back in (loosely) so I could finish the update, booted in to Catalina and swapped the fan connector back to the quiet fan, which keeps the server CPU (Xeon 4-core/8 thread) running at around 22 deg/C.
HP ML10 Gen9 server - 10.15.7 Security Update 2021-002 Catalina