A persistence diagram (PD) summarizes the persistent homology of a filtration. I introduce the concept of a persistence diagram bundle, which is the space of PDs associated with a fibered filtration function (a set $\{f_t: \mathcal{K}^t \to \mathbb{R}\}_{t \in \mathcal{T}}$ of filtrations parameterized by a topological space $\mathcal{T}$). Special cases include vineyards, the persistent homology transform, and fibered barcodes of multiparameter persistence modules. I prove that if $\mathcal{T}$ is a compact $n$-dimensional manifold, then for generic fibered filtration functions, $\mathcal{T}$ is stratified such that within each $n$-dimensional stratum $S$, there is a single PD "template" (a list of birth and death simplices) that can be used to obtain $PD(f_t)$ for any $t \in S$. I also show that not every local section can be extended to a global section. Consequently, the points in the PDs do not typically trace out separate manifolds as $t \in \mathcal{T}$ varies; this is unlike a vineyard, in which the points in the PDs trace out curves ("vines").