Mesh Clipping / Cross Sections

I've seen a few examples of creating shaders to get a "cross section" effect for objects, as seen in these examples on the Unity Forums:

Shader for Clipping Plane
Simple Cross Section Shader

These shaders affect all objects that have the shader attached to them, however, and I need some way to have a different cross section on different instances of the same mesh (generated at runtime). I've thought about creating a separate instance of a material for each object (and then perhaps replacing the material once the effect is no longer needed), but this seems like it might not be the best idea.

Is there a simple way to get a cross section of a mesh (with or without a shader)? I don't necessarily need the "cut off" sections of the mesh to be capped, so a simple clipping plane is all I need, but I don't know if there's a way to make arbitrary clipping planes and move them around. I also need to be able to change how much of a mesh is "cut off" by moving a clipping plane or adjusting shader values (or something similar) at runtime. For my particular problem I just need everything above a certain point to be transparent - only put the material on the mesh up to a certain point. So if there's a way to dynamically chop the mesh off at a certain point, that would be fine.