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The chirp spread spectrum (CSS) modulation scheme is employed by the physical layer of the Long Range (LoRa) communication technology. In this paper, we examine the performance of CSS over time-varying channels whose gain may change during the reception of a LoRa frame. This is in contrast to the usually employed model in the literature, which assumes the channel gain to be constant throughout a frame. Specifically, we investigate the effects of exponentially correlated Rayleigh fading on the frame-error rate of a CSS receiver in which the channel gain is estimated at the beginning of each frame. Our primary observation is that over rapidly-varying channels, the robustness benefits of the larger spreading factors tend to disappear as the payload size grows. This observation, which is contrary to the common perception that higher spreading factors necessarily provide greater immunity against noise, highlights the need to consider channel characteristics and payload sizes in allocating the spreading factor for reliable and energy-efficient LoRa communications.