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We present a formalism for computing time-dependent fluctuations in the cosmological photoionizing radiation background, extending background fluctuation models beyond the steady-state approximation. We apply this formalism to estimate fluctuations in the H I Ly α flux redshift-space power spectrum and its spatial correlation function at redshifts 2 < z < 4, assuming the photoionization background is dominated by quasi-stellar objects (QSOs) and/or galaxies. We show that the shot noise in the power spectrum due to discrete sources is strongly suppressed relative to the steady-state value at low wavenumbers by a factor proportional to the lifetime of the sources, and that this suppression may be used to constrain QSO lifetimes. The total H I Ly α power spectrum including shot noise is affected at tens of per cent on short scales and by as much as an order of magnitude or more on scales exceeding the mean free path. The spatial correlation function is similarly found to be sensitive to the shot noise, although moderately insensitive to the effects of time dependence on the non-shot-noise contribution. Photoionization rate fluctuations substantially modify the shape of the baryonic acoustic oscillation peak in the correlation function, including a small increase in its position that must be accounted for to avoid biasing estimates of cosmological parameters based on the peak position. We briefly investigate solving the full frequency-dependent equation, finding that it agrees with the frequency-independent to better than per cent accuracy. Simple formulas are provided for the power spectrum of fluctuations in the photoionization rate that approximate the full computations.