Hello schveiguy2,
Welcome to Apple Support Communities!
If we understand your post correctly, since upgrading to Big Sur the search field in Apple Mail is slow to respond. We're happy to help!
Let's try an SMC reset.
How to reset the SMC of your Mac
You may also want to try an NVRAM reset.
Reset NVRAM or PRAM on your Mac
We also recommend starting in safe mode. Sometimes just starting in this mode and then restarting can resolve issues like this: How to use safe mode on your Mac
You can also create another user on your Mac and test to see if you experience the same issue.
Set up users, guests, and groups on Mac
Let us know if you have any questions.
Thank you for using Apple Support Communities.
Take care!