Some new emoji missing or broken from system font

For a couple of months now, since one of the larger emoji addition updates made to iOS/Mac OS X toward the end of 2015, my computer has been showing many of them as missing or broken. I'm sure this isn't normal, but I haven't found any remedy or even anyone else who has had this issue.

If possible, I'd love to try and fix it without reinstalling OS X from scratch. I'm not sure if anyone is aware of a terminal command that could uninstall and reinstall the OS X emoji library, or some other method of updating this.

The issue is found all across the system, not just in Messages or Mail. Some emoji in the list are replaced by a black stroked rectangle, while others seem to be just completely missing.

My software is completely up to date, of course, and this issue has persisted through multiple 10.11.x updates.

Any insight would be great. Attached are screenshots of the issue.

Thank you!