This paper describes a blind source separation method for multichannel audio signals, called NF-FastMNMF, based on the integration of the normalizing flow (NF) into the multichannel nonnegative matrix factorization with jointly-diagonalizable spatial covariance matrices, a.k.a. FastMNMF. Whereas the NF of flow-based independent vector analysis, called NF-IVA, acts as the demixing matrices to transform an M-channel mixture into M independent sources, the NF of NF-FastMNMF acts as the diagonalization matrices to transform an M- channel mixture into a spatially-independent M-channel mixture represented as a weighted sum of N source images. This diagonalization enables the NF, which has been used only for determined separation because of its bijective nature, to be applicable to non-determined separation. NF-FastMNMF has time-varying diagonalization matrices that are potentially better at handling dynamical data variation than the time-invariant ones in FastMNMF. To have an NF with richer expression capability, the dimension-wise scalings using diagonal matrices originally used in NF-IVA are replaced with linear transformations using upper triangular matrices; in both cases, the diagonal and upper triangular matrices are estimated by neural networks. The evaluation shows that NF-FastMNMF performs well for both determined and non-determined separations of multiple speech utterances by stationary or non-stationary speakers from a noisy reverberant mixture.