Accurate forecasting of the grid carbon intensity factor (CIF) is critical for enabling demand-side management and reducing emissions in modern electricity systems. Leveraging multiple interrelated time series, CIF prediction is typically formulated as a multivariate time series forecasting problem. Despite advances in deep learning-based methods, it remains challenging to capture the fine-grained local-temporal dependencies, dynamic higher-order cross-variable dependencies, and complex multi-frequency patterns for CIF forecasting. To address these issues, we propose a novel model that integrates two parallel modules: 1) one enhances the extraction of local-temporal dependencies under multi-frequency by applying multiple wavelet-based convolutional kernels to overlapping patches of varying lengths; 2) the other captures dynamic cross-variable dependencies under multi-frequency to model how inter-variable relationships evolve across the time-frequency domain. Evaluations on four representative electricity markets from Australia, featuring varying levels of renewable penetration, demonstrate that the proposed method outperforms the state-of-the-art models. An ablation study further validates the complementary benefits of the two proposed modules. Designed with built-in interpretability, the proposed model also enables better understanding of its predictive behavior, as shown in a case study where it adaptively shifts attention to relevant variables and time intervals during a disruptive event.