Recurrent stochastic configuration networks (RSCNs) are a class of randomized learner models that have shown promise in modelling nonlinear dynamics. In many fields, however, the data generated by industry systems often exhibits nonstationary characteristics, leading to the built model performing well on the training data but struggling with the newly arriving data. This paper aims at developing a self-organizing version of RSCNs, termed as SORSCNs, to enhance the continuous learning ability of the network for modelling nonstationary data. SORSCNs can autonomously adjust the network parameters and reservoir structure according to the data streams acquired in real-time. The output weights are updated online using the projection algorithm, while the network structure is dynamically adjusted in the light of the recurrent stochastic configuration algorithm and an improved sensitivity analysis. Comprehensive comparisons among the echo state network (ESN), online self-learning stochastic configuration network (OSL-SCN), self-organizing modular ESN (SOMESN), RSCN, and SORSCN are carried out. Experimental results clearly demonstrate that the proposed SORSCNs outperform other models with sound generalization, indicating great potential in modelling nonlinear systems with nonstationary dynamics.