We first consider a class of stage-structured differential equations. By using the time delay as a bifurcation parameter, we analytically prove that these local Hopf bifurcation values are neatly paired, and each pair is jointed by a bounded global Hopf branch. We use the well-known Mackey–Glass equation with a stage structure as an illustrative example to demonstrate that bounded global Hopf branches can induce interesting and rich dynamics. We then study the dynamics of a delayed diffusive hematopoiesis model with Dirichlet boundary conditions. we show that the only positive steady state is a constant solution. By using the delay as a bifurcation parameter, we show that the model has infinite number of Hopf bifurcation values and the global Hopf branches bifurcated from these values are unbounded, which indicates the global existence of periodic solutions.