Lithium-ion (Li-ion) batteries are the major type of battery used in a variety of everyday applications, including electric vehicles (EVs), mobile devices, and energy storage systems. Predicting the Remaining Useful Life (RUL) of Li-ion batteries is crucial for ensuring their reliability, safety, and cost-effectiveness in battery-powered systems. The materials used for the battery cathodes and their designs play a significant role in determining the degradation rates and RUL, as they lead to distinct electrochemical reactions. Unfortunately, RUL prediction models often overlook the cathode materials and designs to simplify the model-building process, ignoring the effects of these electrochemical reactions. Other reasons are that specifications related to battery materials may not always be readily available, and a battery might consist of a mix of different materials. As a result, the predictive models that are developed often lack generalizability. To tackle these challenges, this paper proposes a novel material-based Mixture-of-Experts (MoE) approach for predicting the RUL of batteries, specifically addressing the complexities associated with heterogeneous battery chemistries. The MoE is integrated into a probabilistic framework, called Multiple Non-crossing Quantile Mixture-of-Experts for Probabilistic Prediction (RUL-QMoE), which accommodates battery operational conditions and enables uncertainty quantification. The RUL-QMoE model integrates specialized expert networks for five battery types: LFP, NCA, NMC, LCO, and NMC-LCO, within a gating mechanism that dynamically assigns relevance based on the battery's input features. Furthermore, by leveraging non-crossing quantile regression, the proposed RUL-QMoE produces coherent and interpretable predictive distributions of the battery's RUL, enabling robust uncertainty quantification in the battery's RUL prediction. Trained on seven real-world datasets, the proposed RUL-QMoE achieves strong predictive performance across all battery types, with MAE = 65 (cycles), MAPE = 9.59%, RMSE = 100 (cycles), and R2=96.84%. Compared to traditional models like XGBoost, Random Forest, CNN, and LSTM, the proposed RUL-QMoE model consistently delivers lower RMSE and superior probabilistic insights, including survival probabilities and prediction intervals. The model has been integrated into our Battery AI platform in collaboration with Toyota Motor Engineering & Manufacturing North America, Inc., as part of a broader Battery Foundation Model initiative. This RUL-QMoE model will serve future Toyota EVs' users and battery system designers.