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Beamforming design for intelligent reflecting surface (IRS)-assisted multi-user communication (IRS-MUC) systems critically depends on the acquisition of accurate channel state information (CSI). However, channel estimation (CE) in IRS-MUC systems causes a large signaling overhead for training due to the large number of IRS elements. In this paper, taking into account user mobility, we adopt a deep learning (DL) approach to implicitly learn the historical line-of-sight (LoS) channel features and predict the IRS phase shifts to be adopted for the next time slot for maximization of the weighted sum-rate (WSR) of the IRS-MUC system. With the proposed predictive approach, we can avoid full-scale CSI estimation and facilitate low-dimensional CE for transmit beamforming design such that the signaling overhead is reduced by a scale of $\frac{1}{N}$, where $N$ is the number of IRS elements. To this end, we first develop a universal DL-based predictive beamforming (DLPB) framework featuring a two-stage predictive-instantaneous beamforming mechanism. As a realization of the developed framework, a location-aware convolutional long short-term memory (CLSTM) graph neural network (GNN) is developed to facilitate effective predictive beamforming at the IRS, where a CLSTM module is first adopted to exploit the spatial and temporal features of the considered channels and a GNN is then applied to empower the designed neural network with high scalability and generalizability. Furthermore, in the second stage, based on the predicted IRS phase shifts, an instantaneous CSI-aware fully-connected neural network is designed to optimize the transmit beamforming at the access point. Simulation results demonstrate that the proposed framework not only achieves a better WSR performance and requires a lower CE overhead compared with state-of-the-art benchmarks, but also is highly scalable in the numbers of users.