The propagation of small-scale internal gravity wave packets through a background of large-scale internal gravity waves is investigated using Eikonal theory. When background variations are sufficiently slow, it is possible to ignore both the time dependency and the vertical velocity of the background while retaining the Doppler effects of the horizontal winds. This approximation, here referred to as the low-frequency approximation (LFA), is used as a reference approximation to elucidate the peculiar aspects of wave propagation through time-dependent wave backgrounds. The greatest difference between stationary and time-dependent wave background occurs near levels where the frequency measured in the Earth-fixed frame is canceled by the Doppler term due to the horizontal background wind (LFA critical level). In stationary flow, propagation is blocked, and wave amplitude can become very large. The focus of this paper is non-LFA effects near LFA critical levels (LFA\CLs). Major findings are as follows: the LFA overpredicts variations in the intrinsic frequency in a time-dependent background wave field because the variation of observed frequency is exactly out of phase with the variation of the Doppler term. Non-LFA effects at an LFA\CL are strong because as a packet approaches an LFA\CL, its intrinsic frequency decreases, its ascent slows, and its wavenumber vector rotates toward the vertical and increases in magnitude; thus the packet increasingly senses both the time dependency and the vertical component of the background wind. The maximum cancelation between wave frequency and the Doppler term (i.e., minimum intrinsic frequency) will usually occur above LFA\CL. The minimum intrinsic frequency is significantly nonzero and is not a strong function of vertical wavelength. The background vertical velocity allows freer propagation through intrinsic frequency minima. Counterintuitively, the time variation of wave action near such levels appears less pronounced for the slower waves (the ones that should feel Doppler effects most strongly). Eikonal simulations suggest that intrinsic frequency minima associated with LFA\CL and strong Doppler effects are favored locations for wave breakdown, but wave breakdown in a time-dependent background depends on the initial wave-packet amplitude; thus Doppler effects and time dependency should both play a role in shaping the vertical wavenumber spectrum <Hines, 1991a, b; Broutman et al., 1997>. This work supports findings of Eckermann <1997> that the effects of time dependency can drastically reduce the response of a wave packet to a wave background comprising downward propagating waveforms (upward energy transfer).