In this paper, we investigate the impact of channel aging on the performance of cell-free (CF) massive multiple-input multiple-output (MIMO) systems with pilot contamination. To take into account the channel aging effect due to user mobility, we first compute a channel estimate. We use it to derive novel closed-form expressions for the uplink spectral efficiency (SE) of CF massive MIMO systems with large-scale fading decoding and matched-filter receiver cooperation. The performance of a small-cell system is derived for comparison. It is found that CF massive MIMO systems achieve higher 95%-likely uplink SE in both low-and high-mobility conditions, and CF massive MIMO is more robust to channel aging. Fractional power control (FPC) is considered to compensate to limit the inter-user interference. The results show that, compared with full power transmission, the benefits of FPC are gradually weakened as the channel aging grows stronger.