Large-scale magnetic fields in planets, stars and galaxies are maintained by hydromagnetic dynamo action. A magnetic field builds up due to an appropriate motion of a conducting fluid and saturates with increasing field strength owing to the back reaction of the Lorentz force on the flow. In this study we focus on the latter and address the question under which conditions a saturated velocity field stemming from geodynamo simulations leads to an exponential growth of the magnetic field in a corresponding kinematic calculation. In order to settle this question, we perform global self-consistent geodynamo simulations and calculate the evolution of a kinematically advanced tracer field. The self-consistent velocity field enters the induction equation for the tracer field in each time step, but the tracer field does not contribute to the Lorentz force. This experiment has been established by Cattaneo and Tobias (2008) and is closely related to the test-field method (Schrinner et al. 2007). We find two dynamo regimes in which the tracer field either grows exponentially or approaches a state aligned with the actual self-consistent field after a transition period. Both regimes can be distinguished by the Rossby number and coincide with the dipolar and multipolar dynamo regimes identified by Christensen and Aubert (2006). Dipolar dynamos with low Rossby number are kinematically stable whereas the tracer field grows exponentially in the multipolar dynamo regime. This difference in the saturation process for dynamos in both regimes comes along with differences in their time variability. Within our sample of 20 models, solely kinematically unstable dynamos show dipole reversals and large excursions. The complicated time behaviour of these dynamos presumably relates to the alternating growth of several competing dynamo modes. On the contrary, an eigenmode computation suggests that dynamos with low Rossby number are dominated by only one fundamental mode, which is repeatedly quenched and rebuilt. All other modes in this case are clearly subcritical. In this sense, dynamo models in the low Rossby number regime, i.e. fast rotators, exhibit a simple time dependence and their saturation merely results in a fluctuation around their critical state.