I don’t know how Rhino does it, but it’s not exposed in RhinoCommon. Rhino’s geometry is not procedural, you can create a surface by joining faces together but if you modify one face, the adjacent ones will not be updated because there is no such process of internal construction that remakes the surface from the modified face. To achieve this, after transforming a face, you have to remove the adjacent faces and recreate them preserving the topology.
Any linear transformation (like rotating, scaling, reflection…) can be mapped to vectors, where the beginning of the vector is the original state of the points and the end of the vector is the transformed points. So in theory you can use that component to do what you want, but it doesn’t have support for breps.
@DanielPiker developed his own version of something similar with the Mesh Cage Morph component. But the fundamental problem is still there, if you want to transform a part of the geometry and have it affected according to the connected parts, you need to deal with its construction topology.
The example of @maje90 perhaps explains better the idea and limitations of RhinoCommon.