Method to interpolate between 2 sets of meshes in 2D and 3D space

Hello dear JGO

I’m searching for a method to interpolate between meshes in 2D and 3D space, without deforming the meshes in an unexpected way,
like for instance, an interpolated mesh that is too small and the like.

Like in this presentation from David Rosen: http://www.gdcvault.com/play/1020583/Animation-Bootcamp-An-Indie-Approach

I made out different kinds of interpolations like linear and cubic in his presentation, but don’t know how to actually interpolate
without deforming the mesh (again, in an unexpected way).
Also, i guess that there are rules that have to be followed in order to get the math working right.

Could you please point me in the right direction?

Thank you very much in advance!