We derive general relationships between the observed timescale of diffractive interstellar scintillations and the physical velocities of the observer, the source, and the scattering medium. Our treatment applies exclusively to saturated scintillations of point sources in the strong scattering regime. We show how scintillation observations may be combined with other observables (proper motion and dispersion measure) to yield (1) improvements in galactic models for the free-electron density and (2) estimates of the distance and transverse velocity of individual pulsars. We explicitly consider cases of current astrophysical interest, including hypervelocity pulsars too far above the Galactic plane to allow distance estimates from dispersion measures alone. We also briefly consider scintillations of extragalactic sources, including gamma-ray burst sources at great distances from the Galaxy.