ABSTRACT When fast radio burst (FRB) waves propagate through the local (${\lesssim}\rm 1\, pc$) environment of the FRB source, electrons in the plasma undergo large-amplitude oscillations. The finite-amplitude effects cause the effective plasma frequency and cyclotron frequency to be dependent on the wave strength. The dispersion measure and rotation measure should therefore vary slightly from burst to burst for a repeating source, depending on the luminosity and frequency of the individual burst. Furthermore, free–free absorption of strong waves is suppressed due to the accelerated electrons’ reduced energy exchange in Coulomb collisions. This allows bright low-frequency bursts to propagate through an environment that would be optically thick to low-amplitude waves. Given a large sample of bursts from a repeating source, it would be possible to use the deficit of low-frequency and low-luminosity bursts to infer the emission measure of the local intervening plasma and its distance from the source. Information about the local environment will shed light on the nature of FRB sources.