Thank you for posting this topic. I am not sure exactly which results you are comparing because I have not seen the model or a screenshot of the results however I believe the answer is the same for a core as it is for an individual wall. The core is simply the group of walls summarised.
Please see the pdf file attached which explains the difference between wall moments in detail. I have summarised what I have discussed below:
The reason moments are not shown at the base of a meshed wall is because the moment is not translating into the wall as pure bending. In the mid-pier wall, the horizontal force is decomposed as a single horizontal force at the end of a cantilever. In the meshed wall, it starts to behave more like a 3D object, each FE shell will start to “see-saw” resulting in one side lifting and the other side pushing down. This will in turn leave the horizontal forces applied taken as a vertical reaction.
We consider moment as a measure to simply the way internal forces are given in a member. Moments are given to simplify where the internal forces of a member are translating into tension or compression. These effects are often referred to as push and pull effects. To summarise, moments are a way we as engineers use to simplify the internal forces in members, as example of this is where you have a vertically loaded pinned beam. The sagging moment will leave the top of the beam going into compression and the bottom surface subject to internal tensile forces. Similar effects are happening here however the example is taken as a vertical cantilever instead.
I hope this helps explain how the moments are taken in a meshed shear wall. Please let me know if further clarification is needed.