Is there any research on how accurately people can follow a beat?
You might begin your search with the characteristics of Parkinson's disease. This degenerative disorder is precipitated by the death of neurons in the substantia niagra. This brain structure is composed of multiple neurological nuclei. Its function is to connect cortical regions and nuclei together, most significantly with the basal ganglia. The difference between nuclei and cortical brain circuits is important to your question. 

The symptoms of Parkinson's are directly correlated with the function of nuclei in central nervous system: the integration of information. It's the integration of conscious thought, muscle memory, sensory-motor pathways that makes voluntary movement possible. Disruption of this network in Parkinson's degrades one's ability to voluntarily execute a movement.

The cerebellum also controls voluntary movement. As a cortical structure, it is involved more with the formation and maintenance of muscle memory than information integration. Damage here results in the loss of fine motor control as opposed to gross motor control in Parkinson's. 

This problem has a sensory and motor component. It might be difficult to tackle your problem as a whole. Perhaps breaking it down into smaller pieces would shed some light on what you're looking for. 

Here are 2 articles I found relating motor control with the cerebellum and substantia niagra. 

Relating cerebellar damage with learning sequence information

Relating the substantia niagra with interval timing