As a multi-class variant of the classical egalitarian processor-sharing (EPS) discipline, discriminatory processor sharing (DPS) provides a suitable paradigm to model systems where share exists to control the service access of heterogeneous jobs. Although DPS is a more fine-grained scheduling discipline than EPS, the behavior of closed queueing networks (QNs) with DPS remains poorly understood. We propose in this paper a fluid approach to transient and steady-state analysis of closed QNs comprising delay and DPS stations. Our reference model features an arbitrary routing topology, phase-type service time distributions and class switching. The proposed approach has been validated against simulation for both transient and steady-state analysis. Moreover, we introduce a refined method for approximating response time distributions at either station or system level through transient analysis, showing its improvement upon prior art in handling non-exponentially distributed service times.