The time-delay integration (TDI) technique is increasingly used to improve the signal-to-noise ratio (SNR) of remote sensing and imaging by exposing the scene multiple times. Inspired by the principle of TDI, we propose a TDI-like pushbroom multi-slit hyperspectral imaging (MSHSI) approach. In our system, multiple slits are used to significantly improve the throughput of the system, thereby enhancing the sensitivity and SNR through multiple exposures of the same scene during pushbroom scan. Meanwhile, a linear dynamic model for the pushbroom MSHSI is established, where the Kalman filter (KF) is employed to reconstruct the time-varying overlapped spectral images on a single conventional image sensor. Further, we designed and fabricated a customized optical system that can operate in both multi-slit and single slit modes to experimentally verify the feasibility of the proposed method. Experimental results indicate that the developed system improved SNR by a factor of about 7 compared to that of the single slit mode, while demonstrating excellent resolution in both spatial and spectral dimensions.