Recently, the field of computational neuroscience has seen an explosion of the use of trained neural networks (NNs) to model patterns of neural activity. These NN models are typically characterized by tuned interactions between rate 'units' whose dynamics are governed by smooth, continuous differential equations. However, the response of biological single neurons is better described by all-or-none events - spikes - that are triggered in response to the processing of their synaptic input by the complex dynamics of their membrane. One line of research has attempted to resolve this discrepancy by linking the average firing probability of a population of simplified spiking neuron models to rate dynamics similar to those used for NN units. However, challenges remain to account for complex temporal dependencies in the biological single neuron response and for the heterogeneity of synaptic input across the population. Here, we make progress by showing how to derive dynamic rate equations for a population of spiking neurons with multi-timescale adaptation properties - as this was shown to accurately model the response of biological neurons - while they receive time-varying inputs, leading to plausible asynchronous activity in the network. The resulting rate equations yield an insightful segregation of the population's response dynamics into those driven by the mean signal received by the neural population, and those driven by the variance of the input across neurons, with respective timescales that are in agreement with slice experiments. Further, these equations explain how input variability can shape log-normal instantaneous rate distributions across neurons, as observed in vivo. Therefore, we have derived rate equations that explain how single neuron properties shape rich population dynamics. This opens the way to investigating whether this increased complexity relative to vanilla rate equations could provide useful inductive biases when used in NN models trained to solve specific tasks.