Apple won't upgrade my 2010 Mac Pro past OSX 10.12.6; they simply dropped support for the machine. It works fine in general, plenty of power and plenty of resources, but one thing that has aged poorly is 10.12.6's emoji support. More and more often I saw featureless little boxes instead of the emojis people were dropping into chats, forums, slack, and so forth.
Solving the problem was relatively easy. Here's how I did it:
At this point, you have two copies of Apple Color Emoji.ttc, and the correct one (the one that was just installed) is the one that will be used.
That was all there was to it. The soda emoji below is one of those that was just a little box; now it works: