Point cloud comparison
This is termed 3D object retrieval. There are many possible approaches to follow. First, if your meshes are nice, clean and descriptive, you could then consult the academic literature of computer graphics, while computer vision deals with more noisy scenes, partial visibility, non-meshed data. As I am a computer vision oriented person, I would be biased towards such approaches. But keep in mind that Eurographics organizes a workshop on 3d object retrieval every year, called 3DOR. Each year different problems are addressed, but it is very common to find something meeting your requirements.
The following might be good references in the direction you would like to go:


From Low-Cost Depth Sensors to CAD: Cross-Domain 3D Shape Retrieval via Regression Tree Fields

Database-Assisted Object Retrieval for Real-Time 3D Reconstruction

Feature-based Similarity Search in 3D Object Databases



There are many many more works though. For example, here are some, incorporating deep learning:


3DNet: Large-Scale Object Class Recognition from CAD Models

3D Shape Retrieval Using a Single Depth Image from Low-cost Sensors

3D ShapeNets: A Deep Representation for Volumetric Shapes