Apple stopped updating Big Sur nearly a year ago. It is unlikely they will incorporate any recent emoji releases for that system.
If you want to explore possible options, emoji are stored as a system font in /System/Library/Fonts/. The emoji file is Apple Color Emoji.ttc.
You can add fonts to macOS, and if you can find a font file incorporating the latest emoji, you can theoretically install it. That's about as far as my interest takes me, but if you are so motivated here is one possible source. Be advised I haven't even attempted to do anything with it.
https://github.com/GORAlexComp/AppleColorEmojiFont
Then... Install and validate fonts in Font Book on Mac - Apple Support
Let us know if you're successful. Others might appreciate a tutorial. Thanks!