We investigate the impact of chirped driving fields on the dynamics and generation of Kerr cavity breathers and solitons. Synchronous phase and amplitude modulation of the pumping field can be exploited in order to control soliton dynamics. Here we show that using a phase-modulated super-Gaussian pump permits to stabilize the oscillations of breathing solitons. Moreover, our scheme permits to obtain new dynamical attractors, with a prescribed temporal intra-cavity pattern. Straightforward applications are the deterministic generation of optical frequency soliton combs, optical tweezers, and more generally, all-optical manipulation of light pulses.