The cell-free (CF) massive multiple-input multiple-output (MIMO) system is considered a cutting-edge technology in next-generation mobile communication due to its ability to provide high-performance coverage seamlessly and uniformly. This paper aims to mitigate the negative impact of channel aging due to the movement of user equipment in CF massive MIMO systems by utilizing rate-splitting (RS) technology. Taking into account the outdated channel state information, we obtain the achievable sum spectral efficiency (SE) of the RS-assisted CF massive MIMO system, where the private messages can directly adopt conventional maximum ratio, local minimum mean square error (MMSE), and centralized MMSE precoding schemes. Moreover, we propose a bisection-based precoding scheme that maximizes the minimum SE of common messages, which outperforms the superposition-based and random precoding schemes and exhibits strong robustness in complex mobile scenarios. Furthermore, we derive a novel closed-form sum SE expression for the considered system. The results demonstrate that RS technology can mitigate interference in mobile CF massive MIMO systems, improving overall system performance.