You may find this helpful. It describes how soft instructions can be changed.
For Intel processors at least, the term microcode may apply to both the main processor instruction set and the code used by the internal management processor (though this would be more correctly called firmware).
For the processor to update the microcode store it implements specific hard coded instruction that transfer the code through registers to the code storage area. The microcode update instructions only run in ring zero and are defined and stable for a whole processor family.
This all became quite a mess as Intel released separate microcode updates for Spectre/Meltdown protection when using a Windows vs Linux OS.