In this paper we study an N-queue polling model with switchover times. Each of the queues is fed by a non-decreasing Lévy process, which can be different during each of the consecutive periods within the server's cycle. The N-dimensional Lévy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams. For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch. Our analysis heavily relies on establishing a link between fluid (Lévy input) polling systems and multitype Jirina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the branching property for a discipline. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated.