Centrifugal pendulum vibration absorbers are a type of tuned dynamic absorber used for the attenuation of torsional vibrations in rotating and reciprocating machines. They consist of masses that are constrained to move along a specific path relative to the rotating shaft of the machine. Previous analytical studies have considered the performance of multi-absorber systems with a very specific path and of single absorber systems with general paths. In this paper we investigate the performance and dynamic stability of systems comprised of multiple, identical centrifugal pendulum vibration absorbers that have general paths. The study is carried out by considering a scaling of the system parameters based on physically realistic ranges of dimensionless parameters, which allows for application of the method of averaging. It is found that performance is limited by two distinct types of instabilities. In one of these, the systems of absorbers lose their synchronous character, while the other is a classical nonlinear jump behavior that affects all absorbers identically and leads to disastourous results. These results are used to evaluate the performance of systems with common types of absorber paths, namely circles and cycloids. The analytical results are compared against numerical simulations and good agreement is found.