Copy-move forgery detection aims at detecting duplicated regions in a suspected forged image, and deep learning based copy-move forgery detection methods are in the ascendant. These deep learning based methods heavily rely on synthetic training data, and the performance will degrade when facing new tasks. In this paper, we propose a Transformer-style copy-move forgery detection network named as CMFDFormer, and provide a novel PCSD (Pooled Cube and Strip Distillation) continual learning framework to help CMFDFormer handle new tasks. CMFDFormer consists of a MiT (Mix Transformer) backbone network and a PHD (Pluggable Hybrid Decoder) mask prediction network. The MiT backbone network is a Transformer-style network which is adopted on the basis of comprehensive analyses with CNN-style and MLP-style backbones. The PHD network is constructed based on self-correlation computation, hierarchical feature integration, a multi-scale cycle fully-connected block and a mask reconstruction block. The PHD network is applicable to feature extractors of different styles for hierarchical multi-scale information extraction, achieving comparable performance. Last but not least, we propose a PCSD continual learning framework to improve the forgery detectability and avoid catastrophic forgetting when handling new tasks. Our continual learning framework restricts intermediate features from the PHD network, and takes advantage of both cube pooling and strip pooling. Extensive experiments on publicly available datasets demonstrate the good performance of CMFDFormer and the effectiveness of the PCSD continual learning framework.