Previous works concerning active galactic nuclei (AGNs) variability (e.g., Blandford & McKee 1982) have assumed that the emission characteristics of illuminated clouds are purely a function of the instant continuum flux to which they are exposed. This paper shows that this assumption is not necessarily justified and that the history of exposure accounting for "local delays" due to finite cloud equilibrium times can also be relevant. For this reason, a new formalism is developed in this paper for computing the observational properties of models that have local delays. The nature of the nonlinear behavior that results is calculated for some very simple nonlinear cloud line emission models. It is found that the mean response time is a function of the recent average value of the continuum. Linear models fitted to these nonlinear systems respond too slowly when there are low-energy (and generally rapid) changes in the continuum, yet respond too rapidly when there are high- energy (and generally slow) changes in the continuum. As with systems without local delays, the expression for the time-dependent line flux contains an integration over history of the "spatial" response function, which has structure at lags of the light travel times of the emission region. However, the kernel of this integral itself is a function of additional integrations over individual "cloud" response functions that have structure at lags of the equilibrium times of the cloud properties relevant to line emission. In the linear regime, the response can be approximated using a single response function. The integral of this function over lag is not generally equal to the mean flux in the line. Rather, it differs by a factor that is the strength of response for low- frequency continuum excitations or simply the "asymptotic gain," which is unity only in fully linear models. If instantaneous or linear response is incorrectly assumed, local delays and nonlinear response can make a system appear larger than it actually is. These effects are similar to those that beaming can cause. Local delays can also be a source of asymmetry about the peak of the cross-correlation function.