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As a challenging problem, few-shot class-incremental learning (FSCIL) continually learns a sequence of tasks, confronting the dilemma between slow forgetting of old knowledge and fast adaptation to new knowledge. In this paper, we concentrate on this "slow vs. fast" (SvF) dilemma to determine which knowledge components to be updated in a slow fashion or a fast fashion, and thereby balance old-knowledge preservation and new-knowledge adaptation. We propose a multi-grained SvF learning strategy to cope with the SvF dilemma from two different grains: intra-space (within the same feature space) and inter-space (between two different feature spaces). The proposed strategy designs a novel frequency-aware regularization to boost the intra-space SvF capability, and meanwhile develops a new feature space composition operation to enhance the inter-space SvF learning performance. With the multi-grained SvF learning strategy, our method outperforms the state-of-the-art approaches by a large margin.