Combining a chirped laser pulse with a chromatic lens yields a flying focus—a laser focus that moves dynamically in time. This provides control over the propagation of the peak laser intensity within an extended focal region that can be many times larger than the system's Rayleigh length. Any velocity is achievable, including backward relative to the laser propagation direction. Previous simulation results have shown that a laser beam with a flying focus can create a counter-propagating ionization wave and subsequently pump a frequency-downshifted laser via the stimulated Raman scattering instability. Compared to a conventional Raman amplification scheme, several advantages were highlighted, including improved temperature control, plasma uniformity, and precursor growth mitigation. Here, we extend those results to demonstrate additional benefits: (1) the flying focus makes it possible for an unseeded Raman amplifier to produce a short, high-intensity beam; and (2) the flying focus minimizes collisional absorption of the pump, facilitating amplifier operation at higher plasma densities. Preliminary experiments have laid the groundwork for a high-performance plasma-based laser amplifier. The focal spot dynamics were initially confirmed at low intensity. It was subsequently demonstrated that ionization waves of arbitrary velocity can be produced at higher intensity. Here, we show a counter-propagating ionization front moving at approximately the speed of light—the optimal result for a Raman amplifier.