Continuous-wave-driven Kerr nonlinear microresonators give rise to self-organization in terms of dissipative Kerr solitons, which constitute optical frequency combs that can be used to generate low-noise microwave signals. Here, by applying either amplitude or phase modulation to the driving laser we create an intracavity potential trap to discipline the repetition rate of the solitons. We demonstrate that this effect gives rise to a novel spectral purification mechanism of the external microwave signal frequency, leading to reduced phase noise of the output signal. We experimentally observe that the microwave signal generated from disciplined solitons is injection locked by the external drive at long timescales, but exhibits an unexpected suppression of the fast timing jitter. Counterintuitively, this filtering takes place for frequencies that are substantially lower than the cavity decay rate. As a result, while the long timescale stability of the Kerr frequency comb's repetition rate is improved by more than 4 orders of magnitude, the purified microwave signal shows a reduction of the phase noise by 30 dB at offset frequencies above 10 kHz.