Sound field estimation with moving microphones can increase flexibility, decrease measurement time, and reduce equipment constraints compared to using stationary microphones. In this paper a sound field estimation method based on kernel ridge regression (KRR) is proposed for moving microphones. The proposed KRR method is constructed using a discrete time continuous space sound field model based on the discrete Fourier transform and the Herglotz wave function. The proposed method allows for the inclusion of prior knowledge as a regularization penalty, similar to kernel-based methods with stationary microphones, which is novel for moving microphones. Using a directional weighting for the proposed method, the sound field estimates are improved, which is demonstrated on both simulated and real data. Due to the high computational cost of sound field estimation with moving microphones, an approximate KRR method is proposed, using random Fourier features (RFF) to approximate the kernel. The RFF method is shown to decrease computational cost while obtaining less accurate estimates compared to KRR, providing a trade-off between cost and performance.