I downloaded and installed Big Sur on my 2013 MacPro trash can Saturday Nov 14, evening . It is about a 12.8 Gb download, and took about 45 mintures during the dinner hour. It then took about ~45 minutes to install, rebooting several times. The first attempt at installation locked up on a restart step - so I had to finally turn the machine off, and restart it manually, but after that everything seemed to flow properly.
All the native Apple apps seem to work, iMessage particularly works just fine.
I haven't tried Adobe's software yet, as I don't use that on this older device any longer.
Maybe Safari seems a bit snappier.
I did make certain I had two seperate external back up copies of my boot drive before begnning. Never replace an operating system without a reliable 100% funtional back up copy of your existing boot drive. You might find you need it - I didn't this time, but I have needed my back up drive more than once over the years with an OS upgrade.
edit -> I have subsequently added Big Sur to my just recently repaired ( by Apple ) 2019 13" MacBook Air, and it downloaded and installed smoothly and without issues.