By combining the broad learning and a convolutional neural network (CNN), a block-diagonal constrained multi-stage convolutional broad learning (MSCBL-BD) method is proposed for hyperspectral image (HSI) classification. Firstly, as the linear sparse feature extracted by the conventional broad learning method cannot fully characterize the complex spatial-spectral features of HSIs, we replace the linear sparse features in the mapped feature (MF) with the features extracted by the CNN to achieve more complex nonlinear mapping. Then, in the multi-layer mapping process of the CNN, information loss occurs to a certain degree. To this end, the multi-stage convolutional features (MSCFs) extracted by the CNN are expanded to obtain the multi-stage broad features (MSBFs). MSCFs and MSBFs are further spliced to obtain multi-stage convolutional broad features (MSCBFs). Additionally, in order to enhance the mutual independence between MSCBFs, a block diagonal constraint is introduced, and MSCBFs are mapped by a block diagonal matrix, so that each feature is represented linearly only by features of the same stage. Finally, the output layer weights of MSCBL-BD and the desired block-diagonal matrix are solved by the alternating direction method of multipliers. Experimental results on three popular HSI datasets demonstrate the superiority of MSCBL-BD.