Tracking cells in time-lapse videos is an essential technique for monitoring cell population dynamics at a single-cell level. Current methods for cell tracking are developed on videos with mostly single, constant signals and do not detect pivotal events such as cell death. Here, we present TransientTrack, a deep learning-based framework for cell tracking in multi-channel microscopy video data with transient fluorescent signals that fluctuate over time following processes such as the circadian rhythm of cells. By identifying key cellular events - mitosis (cell division) and apoptosis (cell death) our method allows us to build complete trajectories, including cell lineage information. TransientTrack is lightweight and performs matching on cell detection embeddings directly, without the need for quantification of tracking-specific cell features. Furthermore, our approach integrates Transformer Networks, multi-stage matching using all detection boxes, and the interpolation of missing tracklets with the Kalman Filter. This unified framework achieves strong performance across diverse conditions, effectively tracking cells and capturing cell division and death. We demonstrate the use of TransientTrack in an analysis of the efficacy of a chemotherapeutic drug at a single-cell level. The proposed framework could further advance quantitative studies of cancer cell dynamics, enabling detailed characterization of treatment response and resistance mechanisms. The code is available at https://github.com/bozeklab/TransientTrack.