I operate a single covered process on a site owned by a large company.  I manufacture a regulated substance that I pipe to the other company for use in its processes.  At what point do the piping and substance become part of the other company’s stationary source?

The answer will vary.  The company that owns and maintains the piping should probably consider it part of its stationary source.  If, however, there is a point (e.g., a valve or meter) where the receiving company is considered to take ownership of the substance, then you may decide to divide the piping and its contents at that point.